What is the standard deviation of Asset M and of the portfolio equally invested in assets M, N, and O?
Could Sally reduce her total risk even more by using assets M and N only, assets M and O only, or assets N and O only? Use a 50/50 split between the asset pairs, and find the standard deviation of each asset pair.
Please show all of the steps
1. Find the expected return of the equally weighted
portfolio in the three economic states.
Return of Portfolio in Boom = 1/3 (12%) + 1/3 (19%) +
1/3 (2%) = 11.00%
Return of Portfolio in Normal = 1/3 (8%) + 1/3 (11%) + 1/3 (8%) =
9.00%
Return of Portfolio in Recession = 1/3 (2%) + 1/3 (-2%) + 1/3 (12%)
= 4.00% Expected Return Portfolio = 0.30 x (11%) + 0.50 x (9%) +
0.20 x (4%)
E(rp) = 3.3% + 4.5% + 0.8% = 8.6%
2. Find the expected returns of Asset M
Expected Return Asset M = 0.30 x (12%) + 0.50 x (8%) + 0.20 x
(2%)
E(rM) = 3.6% + 4.0% + 0.4% = 8%
Find the standard deviation of Asset M and the
Portfolio.
Standard Deviation of Asset M = [0.30 x (0.12 - 0.08)2 +
0.50 x (0.08 - 0.08)2 + 0.20 x (0.02 -
0.08)2]1/2
= [(0.30 x 0.0016) + (0.20 x 0.0036)]1/2
= (0.00048 + 0.00072)1/2 = 0.00121/2
= 0.0346 or
3.46%
Standard Deviation of Portfolio = [0.30 x (0.11 -
0.086)2 + 0.50 x (0.09 - 0.086)2 + 0.20 x
(0.4 - 0.086)2]1/2
= [(0.30 x 0.0006) + (0.50 x 0.0000) + (0.20 x
0.0021)]1/2
= (0.0002 + 0.0000 + 0.0004)1/2 =
0.00061/2
= 0.0246 or
2.46%
The benefit of the
portfolio over Asset M alone is an increase in return of 0.6% and a
simultaneous reduction in total risk of 1%.
Find the return of each 50/50 portfolio in the three
different states of the economy.
Return of Portfolio M and N in Boom = 1/2 (12%) + 1/2 (19%) =
15.50%
Return of Portfolio M and N in Normal = 1/2 (8%) + 1/2 (11%) =
9.50%
Return of Portfolio M and N in Recession = 1/2 (2%) + 1/2 (-2%) =
0.00%
Return of Portfolio M and O in Boom = 1/2 (12%) + 1/2 (2%) =
7.00%
Return of Portfolio M and O in Normal = 1/2 (8%) + 1/2 (8%) =
8.00%
Return of Portfolio M and O in Recession = 1/2 (2%) + 1/2 (12%) =
7.00%
Return of Portfolio N and O in Boom = 1/2 (19%) + 1/2 (2%) =
10.50%
Return of Portfolio N and O in Normal = 1/2 (11%) + 1/2 (8%) =
9.50%
Return of Portfolio N and O in Recession = 1/2 (-2%) + 1/2 (12%) =
5.00%
Find the expected returns of each individual asset and each
50/50 combination.
Expected Return on Asset M = 0.30 x (12%) + 0.50 x (8%) + 0.20 x
(2%)
E(rM) = 3.6% + 4.0% + 0.4% = 8%
Expected Return on Asset N = 0.30 x (19%) + 0.50 x (11%) + 0.20 x
(-2%)
E(rN) = 5.7% + 5.5% - 0.4% = 10.8%
Expected Return on Asset O = 0.30 x (2%) + 0.50 x (8%) + 0.20 x
(12%)
E(rO) = 0.6% + 4.0% + 2.4% = 7%
Expected Return on Portfolio MN = 0.30 x (15.5%) + 0.50 x (9.5%) +
0.20 x (0%)
E(rMN) = 4.65% + 4.75% + 0.0% = 9.4%
Expected Return on Portfolio MO = 0.30 x (7%) + 0.50 x (8%) + 0.20
x (7%)
E(rMO) = 2.1% + 4.0% + 1.4% = 7.5%
Expected Return on Portfolio NO = 0.30 x (10.5%) + 0.50 x (9.5%) +
0.20 x (5%)
E(rNO) = 3.15% + 4.75% + 1.0% = 8.9%
Find the standard deviation of each asset and each 50/50
portfolio.
Standard Deviation of Asset M = [0.30 x (0.12 - 0.08)2 +
0.50 x (0.08 - 0.08)2 + 0.20 x (0.02 -
0.08)2]1/2
= [(0.30 x 0.0016) + (0.20 x 0.0036)]1/2
= (0.00048 + 0.00072)1/2 = 0.00121/2
= 0.0346 or
3.46%
If Sally chose a 50/50 split between Asset M and O, the benefit is a decrease in total risk to only a half percent (0.5%).
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