Suppose an individual has $$10,000$ to invest between two stocks whose returns are correlated to each other. T

risky stocks are in two companies: Service Inc. $($S$)$ and Manufacturing Co$.($M$)$ whose joint distribution of returr

given in the table below. Note that the returns on the two assets are such that the probability of one stock giving

higher return when the other stock gives a low return is quite high, whereas the probabilities that both stocks give

high return or a low return at the same time is quite low. We will subsequently alter this table for a variety of

scenarios to see if this pattern of joint returns matter in terms of the optimal portfolio.

The individual is risk averse with utility function over wealth given by $u(w)=\sqrt[\phantom{2}]{w}.$

$2.$a$)$ What is the optimal portfolio for this individual if $s$ is the proportion of the $$10,000$ that is invested in stock

and $1-S$ is invested in $M?$ Explain your answer.