Question

There are only two possible states of the economy. State 1 has a 69% chance of occurring. In State 1, Asset A returns 6.25% and Asset B returns 9.25%. In State 2, Asset A returns -3.50% and Asset B returns -6.50%. A portfolio of just these two assets is invested 41% in Asset A (with Asset B comprising the remainder without any negative weights). What is the standard deviation of the portfolio's returns?

Answer 1

Weight of Asset B =1-Weight of Asset A =1-41% =59%
Probability of state 2=1-Probability of State 1 =1-69% =31%
Expected Return in state 1 =Weight of Asset A*Return in State 1+weight of Asset B*return in state 1
=41%*6.25%+59%*9.25% =8.02%
Expected Return in state 2 =Weight of Asset A*Return in State 2+weight of Asset B*return in state 2
=41%*-3.50+59%*-6.50% =-5.27%
Expected return of portfolio =Probability of State 1*Expected Return in state 1+Probability of State 2*Expected Return in state 2
=69%*8.02%+31%*-5.27% =3.90%
Standard Deviation =(69%*(8.02%-3.90%)^2+31%*(-5.27%-3.90%)^2)^0.5 =6.15%