Investments, tenth edition

  The Optimal Risky Portfolio and the Short-Sales Constraint 

 With the input list used by the portfolio manager, the optimal risky portfolio calls for signifi-

cant short positions in the stocks of France and Canada (see column H of   Spreadsheet 7A.3 ). 

In many cases the portfolio manager is prohibited from taking short positions. If so, we need 

to amend the program to preclude short sales. 

 To accomplish this task, we repeat the exercise, but with one change. We add to the 

Solver the following constraint: Each element in the column of portfolio weights, A18–

A24, must be greater than or equal to zero. You should try to produce the short-sale con-

strained efficient frontier in your own spreadsheet. The graph of the constrained frontier is 

also  shown  in   Figure  7A.2 .      

 Column C of  Spreadsheet 7B.1  shows scenario rates of return for debt,  D.  In column 

D we add 3% to each scenario return and in column E we multiply each rate by .4. 

The spreadsheet shows how we compute the expected return for columns C, D, and 

E. It is evident that the mean increases by 3% (from .08 to .11) in column D and is 

multiplied by .4 (from .08 to 0.032) in column E. 

 Example  7B.1 

Expected Rates of Return 

bod61671_ch07_205-255.indd   249

bod61671_ch07_205-255.indd   249

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P A R T   I I

  Portfolio Theory and Practice

  Now let’s construct a portfolio that invests a fraction of the investment budget,  w ( D ),  in 

bonds and the fraction  w ( E ) in stocks. The portfolio’s rate of return in each scenario and its 

expected return are given by   

r

P

(i)

5 w

D

r

D

(i)

1 w

E

r

E

(i) 

(7B.3)

E(r

P

)

5 ap(i)3w

D

r

D

(i)

1 w

E

r

E

(i)

4 5 ap(i)w

D

r

D

(i)

1 ap(i)w

E

r

E

(i)

5 w

D

E(r

D

)

1 w

E

E(r

E

) 

The rate of return on the portfolio in each scenario is the weighted average of the com-

ponent rates. The weights are the fractions invested in these assets, that is, the portfolio 

weights. The expected return on the portfolio is the weighted average of the asset means. 

 

 

  Spreadsheet 7B.2  lays out rates of return for both stocks and bonds. Using assumed 

weights of .4 for debt and .6 for equity, the portfolio return in each scenario appears 

in column L. Cell L8 shows the portfolio expected return as .1040, obtained using 

the SUMPRODUCT function, which multiplies each scenario return (column L) by 

the scenario probability (column I) and sums the results.