Investments, tenth edition


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investopedia.com/articles/pf/05/061505.asp for asset allocation guidelines for various 

types of portfolios from conservative to very aggressive. What do you conclude about 

your own risk preferences and the best portfolio type for you? What would you expect to 

happen to your attitude toward risk as you get older? How might your portfolio compo-

sition change?

  SOLUTIONS TO CONCEPT CHECKS 

   1.      a. 

  

The first term will be  



 

 

w



D

w



D

3 s


D

2



 because this is the element in the top corner of 

the matrix  

 

 

(s



D

2



 times the term on the column border ( 

w  

 D 

 

) times the term on the row 



border ( w  

 D 

 ). Applying this rule to each term of the covariance matrix results in the sum 

   w



D

2

s



D

2

w



D

w

E

 Cov(r



E

r



D

)

w



E

w

D

 Cov(r



D

r



E

)

w



E

2

s



E

2

,  which is the same as Equation 7.3, 



because Cov( r  

 E 

 ,  r  

 D 

 )  5  Cov( r  

 D 

 ,  r  

  

 ).  

   b.   The bordered covariance matrix is    

w

X

w

Y

w

Z

w

X

s

2



x

Cov(r



X

r



Y

)

Cov(r



X

r



Z

)

w



Y

Cov(r



Y

r



X

)

s



2

Y

Cov(r



Y

r



Z

)

w



Z

Cov(r



Z

r



X

)

Cov(r



Z

r



Y

)

s



2

Z

There are nine terms in the covariance matrix. Portfolio variance is calculated from these nine 

terms:  

 s

P

2

w



X

2

  



s

X

2

w



Y

2

s



Y

2

w



Z

2

 



s

Z

2

 



w

X

w

Y

 

Cov(r



X

 

r



Y

)

w



Y

  

w



X

 

Cov(r



Y

 

r



X

)

 



w

X

 

w



Z

 

Cov(r



X

 

r



Z

)

w



Z

 

w



X

 

Cov(r



Z

 

r



X

)

 



w

Y

 

w



Z

 Cov


 

(r



Y

 

r



Z

)

w



Z

 

w



Y

  

Cov(r



Z

 

r



Y

)

 



w

X

2

s



X

2

w



Y

2

s



Y

2

w



Z

2

s



Z

2

 



1 2w

X

 

w



Y

 Cov(r



X

 

r



Y

)

1 2w



X

 

w



Z

 

Cov(r



X

 

r



Z

)

1 2w



Y

 

w



Z

 

Cov(r



Y

 

r



Z

)

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242 

P A R T   I I



  Portfolio Theory and Practice

   2.  The parameters of the opportunity set are  E ( r  

 D 

 )  5  8%,  E ( r  

 E 

 )  5  13%,  s  

 D 

   5  12%,  s  

 E 

   5  20%, 

and  r ( D,  E )  5  .25. From the standard deviations and the correlation coefficient we generate the 

covariance matrix: 

Fund

D

E

D

144


60

E

60

400



    The  global minimum-variance  portfolio is constructed so that   

 w



D

5

s



E

2

2 Cov(r



D

r



E

)

s



D

2

1 s



E

2

2 2 Cov(r



D

r



E

)

 



5

400


2 60

(144


1 400) 2 (2 3 60)

5 .8019


 w

E

5 1 2 w



D

5 .1981  

    Its expected return and standard deviation are  

E(r

P

) 5 (.8019 3 8) 1 (.1981 3 13) 5 8.99%

 s

P

5 3w



D

2

s



D

2

w



E

2

s



E

2

1 2w



D

w

E

 Cov (r



D

r



E

)

4



1/2

 

5 3(.8019



2

3 144) 1 (.1981

2

3 400) 1 (2 3 .8019 3 .1981 3 60)4



1/2

 

5 11.29%



     For the other points we simply increase  w  

 D 

  from .10 to .90 in increments of .10; accordingly, 

 w  

 E 

  ranges from .90 to .10 in the same increments. We substitute these portfolio proportions in 

the formulas for expected return and standard deviation. Note that when  w  

 E 

   5  1.0, the portfolio 

parameters equal those of the stock fund; when  w  

 D 

   5  1, the portfolio parameters equal those of 

the debt fund. 

     We then generate the following table: 



w

E

w

D

E(r)

   s


0.0

1.0


8.0

12.00


0.1

0.9


8.5

11.46


0.2

0.8


9.0

11.29


0.3

0.7


9.5

11.48


0.4

0.6


10.0

12.03


0.5

0.5


10.5

12.88


0.6

0.4


11.0

13.99


0.7

0.3


11.5

15.30


0.8

0.2


12.0

16.76


0.9

0.1


12.5

18.34


1.0

0.0


13.0

20.00


0.1981

0.8019


  8.99

11.29 minimum variance portfolio

     You can now draw your graph.  

   3.      a.     The computations of the opportunity set of the stock and risky bond funds are like those of 

Question 2 and will not be shown here. You should perform these computations, however, in 

order to give a graphical solution to part  a.  Note that the covariance between the funds is   

Cov(r

A

r



B

) 5 r(A, B) 3 s



A

 3 s


B

5 2.2 3 20 3 60 5 2240   



   b.   The proportions in the optimal risky portfolio are given by   

w

A

5

(10



2 5)60

2

2 (30 2 5)(2240)



(10

2 5)60


2

1 (30 2 5)20

2

2 30(2240)



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  C H A P T E R  

7

  Optimal Risky Portfolios 



243

5 .6818


w

B

 5 1 2 w



A

 5 .3182  

 

 The expected return and standard deviation of the optimal risky portfolio are   



E(r

P

) 5 (.6818 3 10) 1 (.3182 3 30) 5 16.36%

s

P

 5 {(.6818

2

 3 20


2

) 1 (.3182

2

 3 60


2

) 1 [2 3 .6818 3 .3182(2240)]}

1/2

5 21.13%  



 

 Note that portfolio  P  is not the global minimum-variance portfolio. The proportions of the 

latter are given by   

 w



A

5

60



2

2 (2240)


60

2

1 20



2

2 2(2240)

5 .8571

w

B

 5 1 2 w



A

 5 .1429  

 

 With these proportions, the standard deviation of the minimum-variance portfolio is   



 s(min)

5 (.8571


2

3 20


2

)

1 (.1429



2

3 60


2

)

1 32 3 .8571 3 .1429 3 (2240)4



1/2

5 17.57%  

 

 which is less than that of the optimal risky portfolio.  



   c.   The CAL is the line from the risk-free rate through the optimal risky portfolio. This line 

represents all efficient portfolios that combine T-bills with the optimal risky portfolio. The 

slope of the CAL is   

S

5

E(r



P

)

r



f

s

P

5

16.36


2 5

21.13


5 .5376   

   d.   Given a degree of risk aversion,  A,  an investor will choose a proportion,  y,  in the optimal risky 

portfolio of (remember to express returns as decimals when using  A ):   



y

5

E(r



P

)

r



f

As

P

2

5



.1636

2 .05


5

3 .2113


2

5 .5089 


 

This means that the optimal risky portfolio, with the given data, is attractive enough for an 

investor with  A   5  5 to invest 50.89% of his or her wealth in it. Because stock  A  makes up 

68.18% of the risky portfolio and stock  B  makes up 31.82%, the investment proportions for 

this investor are   

Stock A: .5089

3 68.18 5 34.70%

Stock B: .5089

3 31.82 5 16.19%

Total


50.89%

      


   4.  Efficient frontiers derived by portfolio managers depend on forecasts of the rates of return on 

various securities and estimates of risk, that is, the covariance matrix. The forecasts themselves 

do not control outcomes. Thus preferring managers with rosier forecasts (northwesterly frontiers) 

is tantamount to rewarding the bearers of good news and punishing the bearers of bad news. 

What we should do is reward bearers of  accurate  news. Thus if you get a glimpse of the frontiers 

(forecasts) of portfolio managers on a regular basis, what you want to do is develop the track 

record of their forecasting accuracy and steer your advisees toward the more accurate forecaster. 

Their portfolio choices will, in the long run, outperform the field.  

   5.  The  parameters  are   E ( r )  5  15,  s   5  60, and the correlation between any pair of stocks is  r   5  .5.

    a.   The portfolio expected return is invariant to the size of the portfolio because all stocks have 

identical expected returns. The standard deviation of a portfolio with  n   5  25  stocks  is   

s

P

5 3s


2

/n

1 r 3 s

2

(n



2 1)/n4

1/2


5 360

2

/25



1 .5 3 60

2

3 24/254



1/2

5 43.27%   

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  APPENDIX A: A Spreadsheet Model for Efficient Diversification 

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244 


P A R T   I I

  Portfolio Theory and Practice



   b.   Because the stocks are identical, efficient portfolios are equally weighted. To obtain a standard 

deviation of 43%, we need to solve for  n:    

43

2

5



60

2

n

1 .5 3

60

2



(n

2 1)


n

1,849n 5 3,600 1 1,800n 2 1,800



n

5

1,800



49

5 36.73  

 

 Thus we need 37 stocks and will come in with volatility slightly under the target.  



   c.  As  

n  gets very large, the variance of an efficient (equally weighted) portfolio diminishes, 

leaving only the variance that comes from the covariances among stocks, that is   

s

P

5 "r 3 s


2

5 ".5 3 60

2

5 42.43%  



 

 Note that with 25 stocks we came within .84% of the systematic risk, that is, the nonsystematic 

risk of a portfolio of 25 stocks is only .84%. With 37 stocks the standard deviation is 43%, of 

which nonsystematic risk is .57%.  



   d.   If the risk-free is 10%, then the risk premium on any size portfolio is 15  2  10  5  5%.  The 

standard deviation of a well-diversified portfolio is (practically) 42.43%; hence the slope of 

the CAL is   

S 5 5/42.43 5  .1178        

  Several software packages can be used to generate the efficient frontier. We will dem-

onstrate the method using Microsoft Excel. Excel is far from the best program for this 

purpose and is limited in the number of assets it can handle, but working through a simple 

portfolio optimizer in Excel can illustrate concretely the nature of the calculations used in 

more sophisticated “black-box” programs. You will find that even in Excel, the computa-

tion of the efficient frontier is fairly easy. 

 We apply the Markowitz portfolio optimization program to a practical problem of inter-

national diversification. We take the perspective of a portfolio manager serving U.S. clients, 

who wishes to construct for the next year an optimal risky portfolio of large stocks in the U.S 

and six developed capital markets (Japan, Germany, U.K., France, Canada, and Australia). 

First we describe the input list: forecasts of risk premiums and the covariance matrix. Next, 

we describe Excel’s Solver, and finally we show the solution to the manager’s problem. 


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