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 the 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 asset​ pairs, and find the standard deviation of each asset pair.

States	Probability	Asset M Return	Asset N Return	Asset O Return
Boom	32%	11%	20%	3%
Normal	51%	9%	13%	9%
Recession	17%	3%	0%	11%

Answer 1

a.

States	Probability	Asset M Return	Asset N Return	Asset O Return	Return on Portfolio [1/3 X (Asset M Return + Asset N Return + Asset O Return)]
Boom	32%	11%	20%	3%	11.33%
Normal	51%	9%	13%	9%	10.33%
Recession	17%	3%	0%	11%	4.67%
Expected return of Asset M [Summation of (Probability X Respective Asset Returns)]	8.62%
Expected return of Asset N	13.03%
Expected return of Asset O	7.42%
Expected return of Portfolio	9.69%
Variance of Asset M [32%*(11%-8.62%)^2+51%*(9%-8.62%)^2+17%*(3%-8.62%)^2]	0.07%
Variance of Asset N	0.44%
Variance of Asset O	0.10%
Variance of Portfolio	0.05%
Std. Dev. Of Asset M	(0.07%)^0.5=2.69%
Std. Dev. Of Asset N	6.66%
Std. Dev. Of Asset O	3.12%
Std. Dev. Of Portfolio	2.32%

The expected return and the risk of investing equally in three assets are 9.69% and 2.32% respectively. If Sally Rogers would have invested in Asset M alone, she would have an expected return of 8.62% with a standard deviation of 2.69%.

b.

States	Probability	Asset M Return	Asset N Return	Asset O Return	Return on M & N Portfolio [1/2 X (Asset M Return + Asset N Return)]	Return on M & O Portfolio	Return on N & O Portfolio
Boom	32%	11%	20%	3%	15.50%	7.00%	11.50%
Normal	51%	9%	13%	9%	11.00%	9.00%	11.00%
Recession	17%	3%	0%	11%	1.50%	7.00%	5.50%
Expected return on M & N Portfolio [Summation of (Probability X Respective portfolio returns)]	10.83%
Expected return on M & O Portfolio	8.02%
Expected return on N & O Portfolio	10.23%
Variance of M & N Portfolio [32%*(15.50%-10.83%)^2+51%*(11%-10.83%)^2+17%*(1.50%-10.83%)^2	0.22%
Variance of M & O Portfolio	0.01%
Variance of N & O Portfolio	0.05%
Std. Dev. of M & N Portfolio (0.22%)^0.5	4.67%
Std. Dev. of M & O Portfolio	1.00%
Std. Dev. of N & O Portfolio	2.15%

Sally can reduce her total risk even more by investing in Assets M & O in equal proportions.