Question

The Minister of Transport released its Festive season road accident statistics which shows that the probability of drivers committing an accident is 8% with utility U(H) = √H, where H stands for year income. The Minister further claims these Festive season road accidents cost the state (in terms of claims lodged) an average of R84 000 per annum. The Road Accident Fund is an insurance scheme providing compulsory indemnity cover to victims of vehicle accidents and taxi drivers. 

(a) Suppose that an average commuter earns R84 000 per annum. What is the expected utility of each commuter if the driver decides not to take insurance.

(b) What is the cost of insurance policy to the Road Accident Fund?       

Answer 1

a)

Given, the utility function:$U=\sqrt{H}$

Probability of accident: $P=8\%$

Cost of the accident: $C=R84000/annum$

Average income of commuter:$H=R84000$

Expected utility without insurance:

$E(U)=P×\sqrt{income\space after \space bearing accident \space cost}+(1-p)×\sqrt{income}\\E(U)=P×\sqrt{income-cost}+(1-p)×\sqrt{income}\\E(U)=P×\sqrt{H-C}+(1-p)×\sqrt{H}\\E(U)=0.08×\sqrt{84000-84000}+(1-0.08)×\sqrt{84000}\\E(U)=0.08×\sqrt{0}+(0=92)×\sqrt{84000}\\E(U)=0+0.92×289.82\\E(U)=266.63$

b) Cost of insurance policy to Road accident fund:

Risk premium

$E(U-P)\ge E(U);$ where p is premium

$\sqrt{H-P}\ge266.63\\\sqrt{84000-P}\ge266.63\\84000-P\ge266.63^3\\84000-P\ge 71091.5569\\P \le 84000-71091.5569\\P\le12908.4431$

The total cost of insurance policy to the accident funds will be the total cost to accident less the premium amount paid by insurance holders.

$Cost\space to \space funds=cost \space of \space accident-premium\\=84000-12908.4431\\=R71091.5569$