Problem 1)
Security
Exp Ret
ST Dev
A
12%
15%
B
20%
45%
Part 1
The question asks for portfolio weights spaced 10% apart, but 5% gives a better
grid for the graph.
Exp Ret
1
0.5
0
0.5
1
0.00%
20.00%
45.00%
45.00%
45.00%
45.00%
45.00%
5.00%
19.60%
42.00%
42.38%
42.76%
43.13%
43.50%
10.00%
19.20%
39.00%
39.77%
40.53%
41.27%
42.00%
15.00%
18.80%
36.00%
37.18%
38.32%
39.42%
40.50%
20.00%
18.40%
33.00%
34.60%
36.12%
37.59%
39.00%
25.00%
18.00%
30.00%
32.04%
33.96%
35.77%
37.50%
30.00%
17.60%
27.00%
29.51%
31.82%
33.97%
36.00%
35.00%
17.20%
24.00%
27.01%
29.72%
32.20%
34.50%
40.00%
16.80%
21.00%
24.56%
27.66%
30.45%
33.00%
45.00%
16.40%
18.00%
22.16%
25.65%
28.73%
31.50%
50.00%
16.00%
15.00%
19.84%
23.72%
27.04%
30.00%
55.00%
15.60%
12.00%
17.64%
21.87%
25.40%
28.50%
60.00%
15.20%
9.00%
15.59%
20.12%
23.81%
27.00%
65.00%
14.80%
6.00%
13.77%
18.52%
22.29%
25.50%
70.00%
14.40%
3.00%
12.28%
17.10%
20.84%
24.00%
75.00%
14.00%
0.00%
11.25%
15.91%
19.49%
22.50%
80.00%
13.60%
3.00%
10.82%
15.00%
18.25%
21.00%
85.00%
13.20%
6.00%
11.05%
14.43%
17.15%
19.50%
90.00%
12.80%
9.00%
11.91%
14.23%
16.22%
18.00%
95.00%
12.40%
12.00%
13.27%
14.43%
15.50%
16.50%
100.00%
12.00%
15.00%
15.00%
15.00%
15.00%
15.00%
150.00%
8.00%
45.00%
38.97%
31.82%
22.50%
0.00%
x
MeanStandard Deviation Frontiers
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Part 2
For perfect negative correlation we have:
Hence,
Hence, the miniminum variance portfolio has 25% invested in asset B and 75% invested in asset A.
You can also read this off the above chart. See also the framed box in the table in part 1.
Part 3
For perfect positive correlation we have:
Hence,
Hence, invest 150% in asset A and 50% in asset B. This requires that you can sell asst B
short. If you cannot, invest 100% in asset A and nothing in asset B.
Part 4
No short sales restrictions:
In both cases (perfect positive and perfect negative correlation) the variance of the
minimum variance portfolio is zero. Expected returns are:
rho = +1
E(R) = 1.5*0.120.5*.20 = 0.08, =
8.00%
rho=1
E(R) = 0.75*0.12+0.25*0.20 =
14.00%
Since the variance of the portfolios that we created is zero and the expected return is higher then
the 5% return earned on the tbill, we are offered an arbitrage opportunity.
We will choose to buy either of the portfolios constructed above and sell the tbill. This arbitrage
portfolio will earn a positive profit with no risk.
Consequently, when the two assets are perfectly positively correlated we will hold asset A long and
short asset B. When the assets are perfectly negatively correlated we will hold both assets long.
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 Fall '09
 Standard Deviation, Variance, Modern portfolio theory, rho

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