Penelope invested 60% of her retirement account in stocks and 40% in gold. Penelope believes that the return to stocks over the next 12 months is normally distributed with mean rate of return of 15% and standard deviation of return of 25%. Penelope also believes that the return to gold over the next 12 months is normally distributed with mean rate of return of 30% and standard deviation of return of 40%. Finally, Penelope believes that the return to stocks and the return to gold are independent. 1. According to Penelope's beliefs, what are the mean rate of return and the standard deviation of return ot her portfolio?
2. What is the probability that Penelope's portfolio will earn at least 12% in the next 12 months?

1) mean = 21, standard deviation σ = 21.9317

2) the probability that Penelope's portfolio will earn at least 12% in the next 12 months is 0.6591

Step-by-step explanation:

 Given the data in the question;

1. According to Penelope's beliefs, what are the mean rate of return and the standard deviation of return of her portfolio?

Mean μ = Return = ( 60% × 15) + ( 40% × 30 ) = 9 + 12 = 21

mean = 21

standard deviation σ = √( (0.6² × 25²) + ( 0.4² × 40² ) )

standard deviation σ = √( (0.36 × 625) + ( 0.16 × 1600 ) )

standard deviation σ = √( 225 + 256 )

standard deviation σ = √481

standard deviation σ = 21.9317

2)  What is the probability that Penelope's portfolio will earn at least 12% in the next 12 months

so we are to find P( X > 12 )

converting into standard normal variable;

⇒ P( Z  > X-μ  / σ  )

= P( Z  > ( 12-21  / 21.9317  ) )

= P( Z  > ( -9 / 21.9317  ) )

= P( Z  > -0.41 )  

FROM z-score table; P( Z  > -0.41 ) is 0.3409

P( X > 12 ) = 1 - 0.3409

P( X > 12 ) = 0.6591

Therefore,  the probability that Penelope's portfolio will earn at least 12% in the next 12 months is 0.6591