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Following is information for the required returns and standard deviations of returns for A, B, and C.

Here are the expected returns and standard deviations for stocks A, B, and C:

Here are the expected returns and standard deviations for stocks A, B, and C:
Stock ri si

A 7.0% 33.11%

B 10.0% 53.85%

C 20.0% 89.44%

  A B C
A 1.0000 0.1571 0.1891
B 0.1571 1.0000 0.1661
C 0.1891 0.1661 1.0000

a. Suppose a portfolio has 30 percent invested in A, 50 percent in B, and 20 percent in C. What are the expected return and standard deviation of the portfolio?

wA = 30%
wB = 50%
wC = 20%


rp =  

Hint: for the portoflio standard deviation, start by creating a table like the one in Section 3.1 for the N-asset case. In fact, begin by creating a table with the products of the weights and standard deviations for each pair of stocks. If you are careful about how you construct the formulas, you can copy them. Then take the results from this intermediate table and multiply them by the correlations above.


A B C



wi = 30% 50% 20%



si = 33.11% 53.85% 89.44%



wi x si =      

wi si wi x si


A 30% 33%        
B 50% 54%        
C 20% 89%        







Now multiply the products of wi x si x wj x sj by the correlations given above to create a table like the one in Section 3.1. A      
B      
C      

b. The partial model lists 66 different combinations of portfolio weights. For each combination of weights, find the required return and standard deviation.

Portoflio # wA  wB  wC  Variance sp  rp 
1 0.0 0.0 1.0      
2 0.0 0.1 0.9      
3 0.0 0.2 0.8      
4 0.0 0.3 0.7      
5 0.0 0.4 0.6      
6 0.0 0.5 0.5      
7 0.0 0.6 0.4      
8 0.0 0.7 0.3      
9 0.0 0.8 0.2      
10 0.0 0.9 0.1      
11 0.0 1.0 0.0      
12 0.1 0.0 0.9      
13 0.1 0.1 0.8      
14 0.1 0.2 0.7      
15 0.1 0.3 0.6      
16 0.1 0.4 0.5      
17 0.1 0.5 0.4      
18 0.1 0.6 0.3      
19 0.1 0.7 0.2      
20 0.1 0.8 0.1      
21 0.1 0.9 0.0      
22 0.2 0.0 0.8      
23 0.2 0.1 0.7      
24 0.2 0.2 0.6      
25 0.2 0.3 0.5      
26 0.2 0.4 0.4      
27 0.2 0.5 0.3      
28 0.2 0.6 0.2      
29 0.2 0.7 0.1      
30 0.2 0.8 0.0      
31 0.3 0.0 0.7      
32 0.3 0.1 0.6      
33 0.3 0.2 0.5      
34 0.3 0.3 0.4      
35 0.3 0.4 0.3      
36 0.3 0.5 0.2      
37 0.3 0.6 0.1      
38 0.3 0.7 0.0      
39 0.4 0.0 0.6      
40 0.4 0.1 0.5      
41 0.4 0.2 0.4      
42 0.4 0.3 0.3      
43 0.4 0.4 0.2      
44 0.4 0.5 0.1      
45 0.4 0.6 0.0      
46 0.5 0.0 0.5      
47 0.5 0.1 0.4      
48 0.5 0.2 0.3      
49 0.5 0.3 0.2      
50 0.5 0.4 0.1      
51 0.5 0.5 0.0      
52 0.6 0.0 0.4      
53 0.6 0.1 0.3      
54 0.6 0.2 0.2      
55 0.6 0.3 0.1      
56 0.6 0.4 0.0      
57 0.7 0.0 0.3      
58 0.7 0.1 0.2      
59 0.7 0.2 0.1      
60 0.7 0.3 0.0      
61 0.8 0.0 0.2      
62 0.8 0.1 0.1      
63 0.8 0.2 0.0      
64 0.9 0.0 0.1      
65 0.9 0.1 0.0      
66 1.0 0.0 0.0      

Hint: Use the formula to calculate the variance for each portfolio and then copy it down. This formula should have six values in it: 1 for Stock A, 1 for Stock B, 1 for Stock C, one for the cross-term of A and B, 1 for the cross-term of A and C, and 1 for the cross term of B and C. The results for portfolio #36 should match your results in part a.

c. The partial model provides a scatter diagram (shown below) showing the required returns and standard deviations calculated above. This provides a visual indicator of the feasible set. If you would like a return of 10.50 percent, what is the smallest standard deviation that you must accept?

For rp = 10.50%, the smallest (and the only) standard deviation is 33.87%; see portfolio #49.

832_What are the expected return and standard deviation.png

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