The certainty equivalent is the certain income or payment in dollars that makes a decision$-$

maker indifferent between taking a risk and taking the certain income. Suppose that an

individual investor plan to invest a risky urban project that has a $50-50$ chance of obtaining or

losing $$5,000$ with a starting investment of $$50,000.$ Then answer and discuss the following

relevant questions $($The quantity results should be kept to three decimal places$).$

a$)$ Let I denote the investment income, and calculate the certain income $($i$.$e$.,$ the certainty

equivalent income$)$ that offers the same utility for this investor in the following three

functions: $U(I)=\sqrt[\phantom{2}]{\text{I}},U(I)=ln(I),U(I)=-\frac{1}{\text{I}},$ where $"$In$"$ denotes natural logarithm

$($For $U(I)=-\frac{1}{\text{I}},$ the value of $U(I)$ should at least be kept to eight decimal places$).(9$

marks$)$

b$)$ Comparing the results in part $($a$),$ what do you conclude about these utility functions in

terms of the investor's risk attitude? $(6$ marks$)$

c$)$ Given the initial investment of $$50,000,$ if this investment project has a $50\%$ chance of

obtaining a $$6,000$ return or incurring a $$2,000$ loss in one year, respectively, calculate

the certainty equivalent yield $($return rate$)$ for all the utility functions given in part $($a$)$

$($For $U(I)=-\frac{1}{\text{I}},$ the value of $U(I)$ should at least be kept to eight decimal places$).(12$

marks$)$

d$)$ Comparing the results in part $($c$),$ what do you conclude about this investor's certainty

equivalent yield with the change in her$/$his risk attitude? $(6$ marks$)$