Self-Insurance

Toyz Limited is a manufacturer of toys for exports. The size of export has been stable in recent years and is expected to stay at the same level in the foreseeable future. In the last 5 years, Toyz had been sued by foreign consumers for product defects once every quarter. All of the lawsuits were settled out- of-court. The table below presents the size of settlement and the corresponding number of lawsuits.

Year 20 Quarters from June 2013 to June 2018

Size of Settlement (US$ millions) $10 $1 $9 $6 $7

Number of Lawsuits 1 5 3 6 5

Toyz is considering the use of liability insurance from Nopay Insurance and self-insurance against future losses due to lawsuits. Toyz's risk manager believes that the size of quarterly settlement will follow the same distribution as in the last 5 years.

(a)Suppose Toyz wants to buy liability insurance from Nopay Insurance to protect against quarterly expected loss. What should be the fair quarterly premium?

(b)Suppose Toyz figures that it can comfortably handle quarterly liability loss up to the expected loss level. On top of that, Toyz is considering setting up a risk fund that will protect Toyz 80% of the time. Determine the size of the risk fund.

(c)Suppose Toyz also considers buying liability insurance to protect against extreme loss that will wipe out the entire risk fund in (b). Determine the fair premium for protection against extreme loss.