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		29​%		13​%		22​%		5%
Normal		53​%		11​%		15​%		11​%
Recession		18%		5​%		2​%		13​%

Answer 1

Expected return (E) = Sum of probability * asset return

E_M = 0.29*0.13 + 0.53*0.11 + 0.18*0.05 = 10.5%

E_N = 0.29*0.22 + 0.53*0.15 + 0.18*0.02 = 14.7%

E_O = 0.29*0.05 + 0.53*0.11 + 0.18*0.13 = 9.6%

A.

When investment is done equally in all the three assets, the returns would be the average for each scenario.

Boom = (13+22+5)/3= 13.3%

Normal = (11+15+11)/3 = 12.3%

Recession = (5+2+13)/3 = 6.6%

E_MNO = 0.29*0.133 + 0.53*0.123 + 0.18*0.066 = 11.5%

Risk(R) is Standard Deviation

R_M = 0.29*(0.13 - 0.105)² + 0.53*(0.11 - 0.105)² + 0.18*(0.05 - 0.105)² = 7.39%

R_MNO= 0.29*(0.133 - 0.115)² + 0.53*(0.123 - 0.115)² + 0.18*(0.066 - 0.115)² = 5.6%

So we can see, the risk in investing only in asset M is more as compared to the risk in investing in all the three assets equally. Also the return is lower for asset M compared to the combined portfolio comprising of the three asset in equal proportions.

B.

Returns for a combination of M and O

Boom = (13+5)/2= 9%

Normal = (11+11)/2 =11%

Recession = (5+13)/2 = 9%

E_MO​​​​​ = 0.29*0.09 + 0.53*0.11 + 0.18*0.09 = 10%

R_MO = 0.29*(0.09-0.1)² + 0.53*(0.11-0.1)² + 0.18*(0.09-0.1)² = 1%

Returns for a combination of M and N

Boom = (13+22)/2 = 17.5%

Normal = (11+15)/2 = 13%

Recession = (5+2)/2 = 3.5%

E_MN = 0.29*0.175 + 0.53*0.13 + 0.18*0.035 = 12.6%

R_MN = 0.29*(0.175-0.126)² + 0.53*(0.13-0.126)² + 0.18*(0.035-0.126)² = 21.95%

Returns for a combination of N and O

Boom = (22+5)/2 = 13.5%

Normal = (15+11)/2 = 13%

Recession = (2+13)/2 = 7.5%

E_NO = 0.29*0.135 + 0.53*0.13 + 0.18*0.075 = 12.1%

R_NO = 0.29*(.135-0.121)² + 0.53*(0.13-0.121)² + 0.18*(0.075-0.121)² = 4.8%

Comparing the risks across the combinations of asset pairs, the combination of asset M and O has the lowest risk.