As a portfolio manager, you$$re benchmarked against the Nasdaq index, which is a passive portfolio. Of the stocks in this passive portfolio, you believe only $2$ are mispriced. You$$ve forecast AAPL to have an alpha of $+5\%($its beta is $1\mathrm{.}5$ and its residual standard deviation is $20\%)$ and META to have an alpha of $+3\%($its beta is $1\mathrm{.}2$ and its residual standard deviation is $25\%).$ The forecast for Nasdaq index$$s expected return is $17\%$ and standard deviation of returns is $22\%.$ Treasuries earn $2\%.$ You have $$1$ to invest. What should be the fraction of each dollar invested in the $3$ assets: Nasdaq, AAPL, and META? Make sure the sum adds up to $1.$

Please show your workings for:

Initial Fraction of AAPL and META in the active portfolio $($given only two securities$,$ you could get whacky numbers here$).$

Alpha of the active portfolio.

Beta of the active portfolio.

Unsystematic variance of the active portfolio

Fraction of the optimal risky portfolio that will be invested in the active portfolio

Final weights of the $3$ components of the risky portfolio