Question

Benefits of diversification. Sally Rogers has decided to invest her wealth equally across the following three assets a. What are her expected returns and the risk from her investment in the three assets How do they compare with investing in asset M alone? Hint: Find t standard deviations of asset M and of the portfolio equally invested in assets M, N, and O b. 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 pairs, and find the standard deviation of each asset pair States Boom Probability Asset M Return Asset N Return Asset O Return 30% 50% 20% 12% 8% 2% 19% 11% 2% 2% 8% 12% a. What is the expected return of investing equally in all three assets M, N, and O? Round to two decimal places.)

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

Answer 1

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(r_p) = 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(r_M) = 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 = 0.0012^1/2
= 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 = 0.0006^1/2
= 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(r_M) = 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(r_N) = 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(r_O) = 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(r_MN) = 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(r_MO) = 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(r_NO) = 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 = 0.0012^1/2
= 0.0346 or 3.46%

Standard Deviation of Asset N [0.30 × (0.19-0.108)? + 0.50 × (0.1 1-0. 108)?+ 0.20 × (-0.02-0. 108)]12 [0.30 x 0.00670.50 x 0.0000 0.20 x 0.0164112 = [0.0020 + 0.0000 + 0.003112 = [0.0053]12 0.0728 or 7.2890 Standard Deviation of Asset O -[0.30x (0.02 0.07)20.50 x (0.08 0.07)2 0.20 x (0.12 = [0.30 × 0.0025 + 0.50 × 0.0001 + 0.20 × 0.0025]12 - [0.0008 0.0001 0.000512 [0.0013]12 0.0712 0.0361 or 3.61% Standard Deviation of Portfolio MN -|0.30x (0.155-0.094)?+ 0.50x (0.095-0.094)2+0.20x (0.0-0.094%]12 - [0.30 x 0.0037+0.50x 0.00000.20x 0.008812 -[0.00110.0000+0.0018 [0.0029]12 0.05.37 or 5.37% Standard Deviation of Portfolio MO - [0.30 x (0.7 0.075)2 0.50x (0.08 0.075) 0.20 x (0.7 0.075)112 [0.30 x 0.0000 0.50 x 0.0000 0.20 x 0.0000112 = [0.0000 + 0.0000 + 0.0000112-[0.0000112 0.0050 or 0.50%

Standard Deviation of Portfolio NO [0.30 × (0.105-0.089)?+ 0.50x (0.095-0.0892 + 0.20 × (0.05-0.089 [0.30ト0.0003 +0.50 × 0.0000 + 0.20 × 0.0015]12 12 = [0.0001 + 0.0000 + 0.000312-[0.000412 = 0.02 or 2%

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%).