现代投资组合理论与投资分析第六章答案

Elton, Gruber, Brown, and Goetzmann

Modern Portfolio Theory and Investment Analysis, 7th Edition

Solutions to Text Problems: Chapter 6

Chapter 6: Problem 1

The simultaneous equations necessary to solve this problem are:

5 = 16Z + 20Z + 40Z

123

7 = 20Z + 100Z + 70Z

123

13 = 40Z + 70Z + 196Z

123

The solution to the above t of equations is:

Z = 0.292831

1

Z = 0.009118

2

Z = 0.003309

3

This results in the following t of weights for the optimum (tangent) portfolio:

X = .95929 (95.929%)

1

X = .02987 (2.987%)

2

梦见金蝉脱壳 X = .01084 (1.084%

3

The optimum portfolio has a mean return of 10.146% and a standard deviation of

4.106%.

Chapter 6: Problem 2

The simultaneous equations necessary to solve this problem are:

11  R = 4Z + 10Z + 4Z

F123

14  R = 10Z + 36Z + 30Z

F123

17  R = 4Z + 30Z + 81Z

F123

6-1

Elton, Gruber, Brown, and Goetzmann

Modern Portfolio Theory and Investment Analysis, 7th Edition

Solutions To Text Problems: Chapter 6

The optimum portfolio solutions using Lintner short sales and the given values for R

F

are:

R = 6% R = 8% R = 10%

FFF

Z 3.510067 1.852348 0.194631

1

1.043624 0.526845 0.010070

Z

2

Z 0.348993 0.214765 0.080537

3

X0.715950 0.714100 0.682350

1

0.212870 0.203100 0.035290

X

2

X 0.711800 0.082790 0.282350

3

Tangent (Optimum) Portfolio

Mean Return 6.105% 6.419% 11.812%

Tangent (Optimum) Portfolio

Standard Deviation 0.737% 0.802% 2.971%

Chapter 6: Problem 3

Since short sales are not allowed, this problem must be solved as a quadratic

programming problem. The formulation of the problem is:

RR

P



F

max



X



P

subject to:



X1

i1



N

i



X0

i

 i

Elton, Gruber, Brown, and Goetzmann

Modern Portfolio Theory and Investment Analysis, 7th Edition

Solutions To Text Problems: Chapter 6

6-2

Chapter 6: Problem 4

This problem is most easily solved using The Investment Portfolio software that

comes with the text, but, since all pairs of asts are assumed to have the same

correlation coefficient of 0.5, the problem can also be solved manually using the

constant correlation form of the Elton, Gruber and Padberg “Simple Techniques”

described in a later chapter.

To u the software, open up the Markowitz module, lect “file” then “new” then

“group constant correlation” to open up a con梦到水是什么意思 stant correlation table. Enter the

input data into the appropriate cells by first double clicking on the cell to make it

active. Once the input data have been entered, click on “optimizer” and then

“run optimizer” (or simply click on the optimizer icon). At that point, you can either

lect “full Markowitz” or “simple method.”

If you lect “full Markowitz,” you then lect “short sales allowed/riskless lending

and borrowing” and then enter 4 for both the lending and borrowing rate and

click “OK.” A graph of the efficient frontier then appears. You may then hit the

“Tab” key to jump to the tangent portfolio, th自我介绍面试模板 en click on “optimizer” and then

“show portfolio” (or simply click on the “show portfolio” icon) to view and print the

composition (investment weights), mean return and standard deviation of the

tangent (optimum) portfolio.

I领会的近义词 f instead you客服工作 lect “simple method,” you then lect “short sales allowed with

riskless ast” and enter 4 for the riskless rate and click “OK.” A table showing the

investment weights of the tangent portfolio then appears.

Regardless of the method ud, the resulting investment weights for the optimum

portfolio are as follows:

Ast i X

i

5.999%

1

17.966%

2

3 21.676%

4 0.478%

29.585%

5

6 12.693%

59.170%

7

14.793%

8

9 3.442%

10 189.224%

Elton, Gruber, Brown, and Goetzmann

Modern Portfolio Theory and Investment Analysis, 7th Edition

Solutions To Text Problems: Chapter 6

6-3

Given the above weights, the opti最新等着我 mum (tangent) portfolio has a mean return of

18.907% and a standard deviation of 3.297%. The efficient frontier is a positively

sloped straight line starting at the riskless rate of 4% and extending through the

tangent portfolio (T) and out to infinity:

Chapter 6: Problem 5

Since the given portfolios, A and B, are on the efficient frontier, the GMV portfolio

can be ob宁波社保网 tained by finding the minimum-risk combination of the two portfolios:

2



BAB



16201



GMV

X

A



22

33616220



ABAB



2



This gives and

R7.33%

GMV



GMV

3.83%

Also, since the two portfolios are on the efficient frontier, the entire efficient frontier

can then be traced by using various combinations of the two portfolios, starting

with the GMV portfolio and moving up along the efficient frontier (increasing the

weight in portfolio A and decreasing the weight in portfolio B). Since X = 1  X

BA

the efficient frontier equations are:

RXR1XR10X81X

PAAABAA



222



PAAABAAAB



X1X2X1X



2

2



36X161X40X1X

AAAA



2

1

GMVGMV

X1X1

BA

3

Elton, Gruber, Brown, and Goetzmann

Modern Portfolio Theory and Investment Analysis, 7th Edition

Solutions To Text Problems: Chapter 6

6-4

Since short sales are allowed, the efficient frontier大二总结 will extend beyond portfolio A

and out toward infinity. The efficient frontier appears as follows:

Elton, Gruber, Brown, and Goetzmann

Modern Portfolio Theory and Investment Analysis, 7th Edition

Solutions To Text Problems: Chapter 6

6-5