Sally Rogers has decided to invest her wealth equally across the following three assets: 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 the standard deviations of asset M and of the portfolio equally invested in assets M, N, and O. What is the expected return of investing equally in all three assets M, N, and O? (round to two decimal places)
States Probability Asset M
Return Asset N Return Asset O Return
Boom 33% 10% 19%
2%
Normal 54% 8% 12%
8%
Recession 13% 2%
-1% 10%
1.Find the expected return of the equally weighted portfolio in the three economic states.
Return of Portfolio in Boom =1/3 (10%) +1/3 (19%) +1/3 (2%) =10.33%
Return of Portfolio in Normal =1/3 (8%) +1/3 (12%) +1/3 (8%) =9.33%
Return of Portfolio in Recession =1/3 (2%) +1/3 (-1%) +1/3 (10%) =3.67%
2. Find the expected returns of Asset M and the Portfolio.
Expected Return Asset M =0.33 ×(10%) +0.54 ×(8%) +0.13 (2%)
E(rM) =3.3% +4.32% +0.26% = 7.88%
Expected Return Portfolio =0.33 ×(10.33%) +0.54 ×(9.33%) +0.13 (3.67%)
E(rp) =3.4% +5.03% +0.47% = 8.498%
3.Find the standard deviation of Asset M and the Portfolio.
Standard Deviation of Asset M =[0.33 ×(0.10 -0.0788)2+0.54 ×(0.08 -0.0788)2+0.13 ×(0.02 -0.0788)2]1/2
=[(0.33 ×0.00044944) +(.54× 0.00000144)+(0.13 ×0.00345744)]1/2
=(0.00059856)1/2=0.00029928 or 2.99%
Standard Deviation of Portfolio =[0.33 ×(0.1033 -0.0849)2+0.54 ×(0.0933 -0.0849)2+0.13 ×(0.367 -0.0849)2]1/2
=[(0.33 ×0.00033856) +(0.54 ×0.00007056) +(0.13 ×0.07958041)]1/2
=(0.0104952805)1/2 = 0.00524764025 or 5.24%
The benefit of the portfolio over Asset M alone is an increase in return of 0.38% and a simultaneous reduction in total risk of 2.25%.
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