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		28​%		12​%		21​%		0​%
Normal		52​%		9​%		14​%		9​%
Recession		20​%		0​%		1​%		12​%

Benedits of diversitication Sally Rogers has decided to invest her wealth equally across the following three assets a. What ane her expected retums and the risk from her investment in the three assets? How do they compare with investing in asset M alone? Hnt Find the standard deviations of asset M and of the portfolo 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 anly or assets N and O enly? Use a 50/50 splt bebween the asset pairs, and find the standard deviation of each asset pakr States Beom Normal Recession A MFen Probability 20% Asset N Rebn Asset O Retrm 0% 21% 14% 12% 525 9 % 9% 20% 0% 12%

Answer 1

Formulas:

Expected return (Er) = sum of (probability*return)

Portfolio return (Pr) = sum of (probability*portfolio return) where portfolio return = average of Asset M return, Asset N return and Asset O return (same for other portfolios comprising of two assets)

Expected standard deviation = {Sum of (Probability*(Asset return - Expected return of the asset)^2)}^(0.5)

Note: There is no direct formula for calculating the expected return and expected standard deviation with probabilities.

Tables:

Notation	P	Mr	Nr	Or	Pr
States	Probability	Asset M Return	Asset N Return	Asset O Return	Portfolio return
Boom	28.00%	12.00%	21.00%	0.00%	11.00%
Normal	52.00%	9.00%	14.00%	9.00%	10.67%
Recession	20.00%	0.00%	1.00%	12.00%	4.33%
Expected return (Er)		8.04%	13.36%	7.08%	9.49%

Notation	P	Mr
States	Probability	Asset M Return	P*(Mr - Er)^2
Boom	28.00%	12.00%	0.000439
Normal	52.00%	9.00%	0.000048
Recession	20.00%	0.00%	0.001293
Expected return (Er)		8.04%
Variance			0.001780
Standard deviation			4.2188%

Notation	P	Nr
States	Probability	Asset N Return	P*(Nr - Er)^2
Boom	28.00%	21.00%	0.001634
Normal	52.00%	14.00%	0.000021
Recession	20.00%	1.00%	0.003055
Expected return (Er)		13.36%
Variance			0.004711
Standard deviation			6.8637%

Notation	P	Or
States	Probability	Asset O Return	P*(Or - Er)^2
Boom	28.00%	0.00%	0.001404
Normal	52.00%	9.00%	0.000192
Recession	20.00%	12.00%	0.000484
Expected return (Er)		7.08%
Variance			0.002079
Standard deviation			4.5600%

Notation	P	Pr
States	Probability	Portfolio return	P*(Pr - Er)^2
Boom	28.00%	11.00%	0.000064
Normal	52.00%	10.67%	0.000072
Recession	20.00%	4.33%	0.000533
Expected return (Er)		9.49%
Variance			0.000668
Standard deviation			2.5839%

Notation	P	Mr	Nr	Pr1
States	Probability	Asset M Return	Asset N Return	Portfolio 1 return	P*(Pr1 - Er)^2
Boom	28.00%	12.00%	21.00%	16.50%	0.000942
Normal	52.00%	9.00%	14.00%	11.50%	0.000033
Recession	20.00%	0.00%	1.00%	0.50%	0.002081
Expected return (Er)		8.04%	13.36%	10.70%
Variance					0.003056
Standard deviation					5.5281%

Notation	P	Mr	Or	Pr2
States	Probability	Asset M Return	Asset O Return	Portfolio 2 return	P*(Pr2 - Er)^2
Boom	28.00%	12.00%	0.00%	6.00%	0.000068
Normal	52.00%	9.00%	9.00%	9.00%	0.000108
Recession	20.00%	0.00%	12.00%	6.00%	0.000049
Expected return (Er)		8.04%	7.08%	7.56%
Variance					0.000225
Standard deviation					1.4988%

Notation	P	Nr	Or	Pr3
States	Probability	Asset N Return	Asset O Return	Portfolio 3 return	P*(Pr3 - Er)^2
Boom	28.00%	21.00%	0.00%	10.50%	0.000002
Normal	52.00%	14.00%	9.00%	11.50%	0.000085
Recession	20.00%	1.00%	12.00%	6.50%	0.000277
Expected return (Er)		13.36%	7.08%	10.22%
Variance					0.000364
Standard deviation					1.9083%

To summarize:

Asset/Portfolio	Expected return	Expected standard deviation
Asset M	8.04%	4.2188%
Asset N	13.36%	6.8637%
Asset O	7.08%	4.5600%
Portfolio of all 3 assets	9.49%	2.5839%
Portfolio of M & N	10.70%	5.5281%
Portfolio of M & O	7.56%	1.4988%
Portfolio of N & O	10.22%	1.9083%

a). Expected return and risk from each asset is given in the table above. The return of the portfolio with all 3 assets is higher than the return from asset M. The portfolio also has lower risk than asset M. Hence, the portfolio is preferable over asset M alone.

b). As can be seen from the table above, the portfolio comprising of N & O assets, has the second lowest risk among all portfolios with one of the highest returns so risk can definitely be reduced by the portfolio of N & O assets compared to the portfolio of all 3 assets.