There is a 20 percent probability the economy will boom, 70 percent probability it will be normal, and a 10 percent probability of a recession. Stock A will return 18 percent in a boom, 11 percent in a normal economy, and lose 10 percent in a recession. Stock B will return 9 percent in boom, 7 percent in a normal economy, and 4 percent in a recession. Stock C will return 6 percent in a boom, 9 percent in a normal economy, and 13 percent in a recession. What is the expected return on a portfolio which is invested 20 percent in Stock A, 50 percent in Stock B, and 30 percent in Stock C

this question is solved taking two steps, lets first calculate the individual expected return per stock based on the probabilities of growing economy:

[tex]E(r)=P_{boom}*R_{boom}+P_{normal}*R_{normal}+P_{recession}*R_{recession}[/tex]

here E(r) represents the expected return of any stock based on the probability of boom, normal and recession in economy, and the return for each one of those states, so applying to this data we have:

Stock A

[tex]E(r)=20\%*18\%+70\%*11\%+10\%*10\%[/tex]

[tex]E(r)=12.3\%[/tex]

Stock B

[tex]E(r)=20\%*9\%+70\%*7\%+10\%*4\%[/tex]

[tex]E(r)=7.1\%[/tex]

Stock C

[tex]E(r)=20\%*6\%+70\%*9\%+10\%*13\%[/tex]

[tex]E(r)=8.3\%[/tex]

now we have to agregate for total portfolio the return, and this can be done using the next formula:

[tex]E(r)_{p}= E(r)_{A}*w_{A}+E(r)_{B}*w_{B} +E(r)_{C}*w_{C}[/tex]

where E(r) (A) for example represents the expected return of A stock and w(A) is the weight of A stock in total portafolio, so we have:

[tex]E(r)_{p}= 12.3\%*20\%+7.1\%*50\%+8.8\%*30\%[/tex]

[tex]E(r)_{p}= 8.65\%[/tex]