Long Question $3(25$ points total, $5$ points each$)$

Let $R_A$ and $R_B$ denote the returns on securities A and $B,$ respectively. The table below shows the potential outcomes and associated probabilities for $R_A$ and $R_B$ under three scenarios $($Good$,$ Normal, and Poor$)$:

a$)$ Calculate the expected returns, $E(R_A)$ and $E(R_B).$

b$)$ Calculate the variances, Var$(R_A)$ and Var$(R_B).$

c$)$ Compute the covariance between the returns of Stock A and Stock B$.$

d$)$ Quantify and comment on the strength of the relationship between the returns of Stocks A and $B.$

e$)$ An investment portfolio $P$ is the combination of securities A and $B.$ The weights of securities A and $B$ in the portfolio are $w$ and $(1-w),$ respectively. Derive the portfolio's expected return and variance as functions of $w.$