hi guys I am new here. our university just started the course so its not certified yet so I am not sure where these questions should go. but I could really use your help as our teacher isnt very good so i have no idea whats going on. Thanks 1. An investor has wealth X and invests a proportion (alpha) in a risky asset that will increase in value by y% (so that an investment of 1 would increase to 1+y) with probability p and fall to zero with probability 1 − p. The amount not invested in the risky asset will neither increase nor decrease in value. (a) If the investor has a utility function U(w) = ln(w) then show that expected utility is maximimized by maximizing lnX + p ln(1 + alpha*y) + (1 − p) ln(1 − alpha) (b) Hence show that expected utility is maximized when alpha = {yp- (1-p)}/{y} What does the numerator represent? (c) Find the proportion of the investor’s wealth that he would be prepared to invest in an asset that doubled in value with probability 5/6 but became worthless with probability 1/6. Q2. Show that the utility function U(w) = pw has (a) decreasing absolute risk aversion. (b) constant relative risk aversion. Q3. An individual and an insurer both have a utility function U(w) = ln(w). The individual has initial wealth 10 and the insurer has initial wealth 100. (a) Calculate the premium P that the individual would be prepared to pay to fully insure a loss of 5 with probability 50%. (b) Show that the premium Q that the insurer requires satisfies the equation Q^2 + 195Q − 500 = 0 (c) Hence find the premium that the insurer requires. (d) Comment on the difference between P and Q. Thanks Greatly appreciate it
Phew! This is too big a query! The expected utility theory is covered in Chapter 5 of the Course Notes and its application to insurance is covered in Chapter 6 of the Course Notes. The questions that you ask are very similar to questions (and solutions) included in these chapters.
Hi Margaret Thanks for the reply. This might be a stupid question but where can you find the course notes.
Hi Margaret, I cant afford these notes, but is it possible if you could check if I am doing it right. Do u first differentiate [ lnx + pln(1+αy) + (1-p)in(1-α)] wrt α and equate to 0 then u get py - 1 + p - yα =0 then u make α the subject then u differentiate again to see if u get negative answer and if u do it means its maximised. but i dont understand the following questions: * when i differentiate again do I use this term: [{py(1+αy)^-1} -{(1-p)(1-α)^-1}] or do I use this term : py -1+ p - yα * also what happens to U(w) = ln (w). ie whats this used for? (b) also I dont have a clue what does the numerator stand for ie part b , because y=value increased by, p = prob of increasing and (1-p) stands for prob of it falling to 0. (c) when it says y = doubled its value does that mean y = 2(1+y) => y = -2? Please help me. Thanks
Hi, Unfortunately we can't help you directly as it would be unfair to students who purchase our materials and come along to our tutorials. But you never know, a generous student may be willing to help you out. Anna
