Problem $1.(5$ points$)$

You are a portfolio manager with $$100,000$ to invest. Today $($at date $t)$ there are two

opportunities to invest, both yielding a stochastic payoff depending the state of the world

to be realized next month $($at date $t+1).$ In particular, the economy will continue on an

upward trajectory with probability $40\%,$ or will turn into recession with probability $60\%.$

For each dollar invested in a project, the distribution of the project's payoffs is given as

follows.

Suppose that you hold a portfolio $W$ with $$30,000$ invested in project $xx,$ and $$70,000$

invested in project $YY.$

What is the expected payoff and the variance of project $xx?$ What is the expected

payoff and the variance of project $YY?$

What is the expected payoff of your portfolio as of today $($date $t)?$ What is the

expected rate of return of your portfolio as of today $($date $t)?$

What about the standard deviation of your portfolio? What about the standard

deviation of the rate of return?

How does the risk of the portfolio compares with the risk individual projects $($i$.$e$.$

investing the entire $$100,000$ in either project $xx$ or $YY)?$ Evaluate in terms of

the rate of return of the portfolio and of each individual project.

Suppose that your preferences are captured by a mean$-$variance utility function, with

a risk aversion coefficient $=1\mathrm{.}8.$ If given a choice to invest the entire amount of

$$100,000$ in a single project, which would you choose: project $xx,$ project $YY,$ or

a risk$-$free project, $F,$ that yields a rate of return of $r_F=4\%?$

You are given an additional $$60,000$ to invest. How do you allocate it between your

original portfolio, $W,$ and the risk$-$free project, $F($if your risk aversion is $=1\mathrm{.}8)?$