Question

EXERCISE 2 Show that two portfolios on the capital allocation line are perfectly correlated EXERCISE 3

Answer 1

Two portfolios on the capital market line have the same slope but different standard deviation.

The Capital allocation line equation is given by

$E(r_{p})=r_{f}+\left ( \frac{E(r_{a})-r_{f}}{\sigma _{a}} \right )\sigma _{p}$

Here, $\left ( \frac{E(r_{a})-r_{f}}{\sigma _{a}} \right )$ is the slope of the CML and is same for both the portfolios

Lets us consider a case of 2-portfolio on a CML with different portfolio standard deviation $\sigma _{a}$

$E(r_{a})$ is the expected return on the risky-part of the portfolio

$r_{f}$ is the risk-free rate = 2%

$\sigma _{a}$ is the standard deviation on risky-part of the portfolio = 0.4

$\sigma _{p1}$ is the standard deviation of portfolio 1 = 0.2

$\sigma _{p2}$ is the standard deviation of portfolio 2 = 0.25

We use excel, to get different expected return on the portfolios $E(r_{p1})$ and $E(r_{p2})$ using the risk-free rate and standard deviations mentioned

E(Rp1)	0.02+0.2*((E(Ra)-0.02)/0.4)
E(Rp2)	0.02+0.25*((E(Ra)-0.02)/0.4)

E(Ra)	E(Rp1)	E(Rp2)
1%	1.500%	1.37500%
2%	2.000%	2.00000%
3%	2.500%	2.62500%
4%	3.000%	3.25000%
5%	3.500%	3.87500%
6%	4.000%	4.50000%
7%	4.500%	5.12500%
8%	5.000%	5.75000%
9%	5.500%	6.37500%
10%	6.000%	7.00000%
11%	6.500%	7.62500%
12%	7.000%	8.25000%
13%	7.500%	8.87500%
14%	8.000%	9.50000%
15%	8.500%	10.12500%
16%	9.000%	10.75000%
17%	9.500%	11.37500%
18%	10.000%	12.00000%
19%	10.500%	12.62500%
20%	11.000%	13.25000%
21%	11.500%	13.87500%
22%	12.000%	14.50000%
23%	12.500%	15.12500%
24%	13.000%	15.75000%
25%	13.500%	16.37500%
26%	14.000%	17.00000%
27%	14.500%	17.62500%
28%	15.000%	18.25000%
29%	15.500%	18.87500%
30%	16.000%	19.50000%
31%	16.500%	20.12500%
32%	17.000%	20.75000%
33%	17.500%	21.37500%
34%	18.000%	22.00000%
35%	18.500%	22.62500%
36%	19.000%	23.25000%
37%	19.500%	23.87500%
38%	20.000%	24.50000%
39%	20.500%	25.12500%
40%	21.000%	25.75000%
41%	21.500%	26.37500%
42%	22.000%	27.00000%
43%	22.500%	27.62500%
44%	23.000%	28.25000%
45%	23.500%	28.87500%

Now, we calculated the correlation for both these portfolios using excel => Data => Data Analysis => Correlation

E F K B E(Ra) ? X Correlation Input Input Range: ок | Cancel Grouped By: C E(Rp1) 1.500% 2.000% 2.500% 3.000% 3.500% 4.000% 4.500% 5.000% 5.500% 6.000% 6.500% 7.000% 7.500% 8.000% 8.500% 9.000% $C$4:$D$490 Columns O Rows D E(Rp2) 1.37500% 2.00000% 2.62500% 3.25000% 3.87500% 4.50000% 5.12500% 5.75000% 6.37500% 7.00000% 7.62500% 8.25000% 8.87500% 9.50000% 10.12500% 10.75000% 7% 8% 9% 10% 11% 12% Help Labels in first row $F$16 Output options O Output Range: O New Worksheet Ply: O New Workbook 14% 15% 16% 20 ron/ roon

We get the correlation matrix as

	E(Rp1)	E(Rp2)
E(Rp1)	1
E(Rp2)	1	1

Since, the correlation between the portfolios is 1, we observe that the two portfolios (on the CML) are perfectly correlated.