Problem $5$: Attitude to risk and return

You total wealth is $$100,000.$ A project can earn $35\%$ or lose $15\%$ with probability $0\mathrm{.}5.$ Your utility of wealth is U$($W$)=$ Log$($W$).$ You wonder how much of your wealth to invest into it$.$

a$)$ What is the $CE of your risky wealth if you invest $100\%$ of your wealth in the project?

EU$($W$)=$

CE $=$

Do you do it$?$

b$)$ You can borrow at $0\%$ interest rate. What is the maximum $Bmax you can borrow without going bankrupt in the down case. Your agent won$$t let you borrow more than $Bmax.

c$)$ Write the simple formula of your $CE as a function of $B$.$ Consider, you pay back your debt at each outcome.

d$)$ How much would you borrow to optimize your EU $($ or $CE$).$ Just do it numerically in R$.$e$)$ In Figure $3$ plot on the same plot, vs the amount borrowed for $B in $[0,$ $Bmax $],1)$ your expected wealth EW$,2)$ the $ standard deviation of your wealth $\backslash$sigma W$,3)$ the $CE of your wealth. So B is on the horizontal axis and the $3$ quantities on the vertical axis. Use a range of values that allow to see all interesting parts of the functions.

Make sure to show the optimum from question $3)$ on the graph.