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.

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using the table below:

States		Probability		Asset M Return		Asset N Return		Asset O Return
Boom		28​%		11​%		21​%		−1​%
Normal		49​%		8​%		13​%		8%
Recession		23​%		−1​%		11​%		11​%

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 expected return of investing in asset M​ alone? ​(Round to two decimal​ places.)

What is the standard deviation of the portfolio that invests equally in all three assets​ M, N, and​ O? ​(Round to two decimal​ places.)

What is the standard deviation of asset​ M? ​(Round to two decimal​ places.)

By investing in the portfolio that invests equally in all three assets​ M, N, and O rather than asset M​ alone, Sally can benefit by increasing her return by ______ and decrease her risk by _________ ?  ​(Round to two decimal​ places.)

b.  

What is the expected return of a portfolio of​ 50% asset M and​ 50% asset​ N? ​(Round to two decimal​ places.)

What is the expected return of a portfolio of​ 50% asset M and​ 50% asset​ O? ​(Round to two decimal​ places.)

What is the expected return of a portfolio of​ 50% asset N and​ 50% asset​ O? (Round to two decimal​ places.)

What is the standard deviation of a portfolio of​ 50% asset M and​ 50% asset​ N? (Round to two decimal​ places.)

What is the standard deviation of a portfolio of​ 50% asset M and​ 50% asset​ O? ​(Round to two decimal​ places.)

What is the standard deviation of a portfolio of​ 50% asset N and​ 50% asset​ O? ​(Round to two decimal​ places.)

c.

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?  ​(Select the best​ response.)

a. not enough info to answer question

b. yes, a portfolio of 50% of asset M and 50% of asset O, could reduce risk to 1.5%

c. no, none of the portfolios using 50-50 split reduce risk

d. yes, a portfolio of 50% of asset M and 50% of asset N, could reduce risk to 1.5%

Answer 1

a.

Formula for investing in M only (R₁)= Summation ( Probability * Return)

Formula for Risk in M only = Square Root ( Sum( Probability *( Return - R₁)^2))

Formula for Portfolio Return = >

w(1) = Wieght of M

w(2) = Wieght of N

w(3) = Wieght of O

R(1) = Return of M

R(2) = Return of N

R(3) = Return of O

SD (1) = S.D. of M

SD (2) = S.D. of N

SD (3) = S.D. of O

COV (1,2) = Covariance of M&N

COV (1,3) = Covariance of M&O

COV (2,3) = Covariance of O&N

Return of Portfolio = Sum ( W(n)* R(n) )

Risk of Portfolio = SQRT ( Sum ( W(n) * S.D. (n)^2) + 2* Sum (W(n1)*W(n2)* S.D.(n1)* S.D. (n2) ) )

Now, the table for the results are as follows:

States	Asset M Return	Asset N Return	Asset O Return
Expected Return	6.77%	14.78%	6.17%
St. Deviation	4.43%	3.96%	4.63%

CoVariance Table	M	N	O
M	0.20%	0.00%	0.00%
N	0.00%	0.16%	0.00%
O	0.00%	0.00%	0.21%

	M	N	O
Weights	33%	33%	33%
Return	9.24%
Risk	4.35%

As seen from the table above,

The Return for Portfolio (M,N,O) > Return from M [ 9.24% > 6.77% ]

Risk of Portfolio (M,N ,O) < Risk of M [4.35 < 4.43 ]

b.

The results are as follows:

	M	N	O
Weights	50%	50%	0%
Return	10.78%
Risk	4.20%

	M	N	O
Weights	0%	50%	50%
Return	10.48%
Risk	4.31%

	M	N	O
Weights	50%	0%	50%
Return	6.47%
Risk	4.53%

As seen Risk (Portfolio (M,N) ) < Risk (Portfolio (O, N) ) < Risk (Portfolio (M,N,O)) < Risk (Portfolio (M,O) )

Answer: Yes Sally can reduce the risk further. The primary reason for the same is that Covariance between all the assets is approximately 0.