Sempronius has a logarithmic utility function. He owns an asset the value of which is 12 ducats. This asset is subject to a potential loss of 8 ducats with probability 1 4.

(a) Which insurance premium should Sempronius pay if he selects β = 1 and if the loading is equal to 0.2?

(b) What is the optimal value of β?

(c) Compute Sempronius’s expected utility at the optimal value of β and compare it with his expected utility when β = 0 or when β = 1.

(d) What happens to β∗

(i) when initial wealth becomes 16 ducats while the characteristics of the potential loss are unchanged? Relate your result to Proposi tion 3.3(i).

(ii) whentheloadingfallstozero?RelateyourresulttoProposition3.1.

(e) Ca¨ ıus has the same utility function as Sempronius but his assets are worth 24 ducats and they are subject to a potential loss of 16 with probability 1 4. Show that the optimal value of β for Ca¨ıus is equal to that of Sempronius. Relate your result to a property of the relative risk aversion coefficient when u is logarithmic. (Hint: alternatively you may show that for any β, the expected utility of Ca¨ıus is equal to ln2 +

the expected utility of Sempronius.)