Question

There are only two possible states of the economy. State 1 has a 77% chance of occurring. In State 1, Asset A returns 5.25% and Asset B returns 8.25%. In State 2, Asset A returns -3.10% and Asset B returns -6.10%. A portfolio of just these two assets is invested 33% 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 =1-33% =67%
Probability of State 2 =1-Probability of state 1=1-77%=23%
Expected Return in state 1=Weight of A*return of A in State 1+Weight of B*return of B in State 1 =33%*5.25%+67%*8.25% =7.26%
Expected Return in state 2=Weight of A*return of A in State 2+Weight of B*return of B in State 2=33%*-3.10%+67%*-6.10% =-5.11%

Expected return of Portfolio return =Probability of State 1*Expected Return in state 1+Probability of State 2*Expected Return in state 2=77%*7.26%+23%*-5.11% =4.4149%

Standard Deviation of the portfolio's returns =(77%*(7.26%-4.4149%)^2+23%*(-5.11%-4.4149%)^2)^0.5 =5.2057% or 5.21%