Insurance and risk mitigation decisions

3 Emma has won a holiday worth \(\pounds 1000\). She is wondering whether or not to take out an insurance policy which will pay out \(\pounds 1000\) if she should fall ill and be unable to go on the holiday. The insurance company tells her that this happens to 1 in 200 people. The insurance policy costs \(\pounds 10\). Thus Emma's monetary value if she buys the insurance and does not fall ill is \(\pounds 990\).

1. Draw a decision tree for Emma's problem. Use the EMV criterion in your calculations.
2. Interpret your tree and say what the maximum cost of the insurance would have to be for Emma to consider buying it if she uses the EMV criterion. Suppose that Emma's utility function is given by utility \(= \sqrt [ 3 ] { \text { monetary value } }\).
3. Using expected utility as the criterion, should Emma purchase the insurance? Under this criterion what is the cost at which she will be indifferent to buying or not buying it? Emma could pay for a blood pressure check to help her to make her decision. Statistics show that \(75 \%\) of checks are positive, and that when a check is positive the chance of missing a holiday through ill heath is 0.001 . However, when a check is negative the chance of cancellation through ill health is 0.017.
4. Draw a decision tree to help Emma decide whether or not to pay for the check. Use EMV, not expected utility, in your calculations and assume that the insurance policy costs \(\pounds 10\). What is the maximum amount that she should pay for the blood pressure check?

1 Keith is wondering whether or not to insure the value of his house against destruction. His friend Georgia has told him that it is a waste of money. Georgia argues that the insurance company sets its premiums (how much it charges for insurance) to take account of the probability of destruction, plus an extra fee for its profit. Georgia argues that house-owners are, on average, simply paying fees to the insurance company. Keith's house is valued at \(\pounds 400000\). The annual premium for insuring its value against destruction is \(\pounds 100\). Past statistics show that the probability of destruction in any one year is 0.0002 .

1. Draw a decision tree to model Keith's decision and the possible outcomes.
2. Compute Keith's EMV and give the course of action which corresponds to that EMV.
3. What would be the insurance premium if there were no fee for the insurance company? For the remainder of the question the insurance premium is still \(\pounds 100\).
   Suppose that, instead of EMV, Keith uses the utility function utility \(= ( \text { money } ) ^ { 0.5 }\).
4. Compute Keith's utility and give his corresponding course of action. Keith suspects that it may be the case that he lives in an area in which the probability of destruction in a given year, \(p\), is not 0.0002 .
5. Draw a decision tree, using the EMV criterion, to model Keith's decision in terms of \(p\), the probability of destruction in the area in which Keith lives.
6. Find the value of \(p\) which would make it worthwhile for Keith to insure his house using the EMV criterion.
7. Explain why Keith may wish to insure even if \(p\) is less than the value which you found in part (vi). [1]
   (a) A national Sunday newspaper runs a "You are the umpire" series, in which questions are posed about whether a batsman in cricket is given "out", and why, or "not out". One Sunday the readers were told that a ball had either hit the bat and then the pad, or had missed the bat and hit the pad; the umpire could not be sure which. The ball had then flown directly to a fielder, who had caught it. The LBW (leg before wicket) rule is complicated. The readers were told that this batsman should be given out (LBW) if the ball had not hit the bat. On the other hand, if the ball had hit the bat, then he should be given out (caught). Readers were asked what the decision should be. The answer given in the newspaper was that this batsman should be given not out because the umpire could not be sure that the batsman was out (LBW), and could not be sure that he was out (caught).

1 Martin is considering paying for a vaccination against a disease. If he catches the disease he would not be able to work and would lose \(\pounds 900\) in income because he would have to stay at home recovering. The vaccination costs \(\pounds 20\). The vaccination would reduce his risk of catching the disease during the year from 0.02 to 0.001 .

1. Draw a decision tree for Martin.
2. Evaluate the EMV of Martin's loss at each node of your tree, and give the action that Martin should take to minimise the EMV of his loss. Martin can answer a medical questionnaire which will give an estimate of his susceptibility to the disease. If he is found to be susceptible, then his chance of catching the disease is 0.05 . Vaccination will reduce that to 0.0025 . If he is found not to be susceptible, then his chance of catching the disease is 0.01 and vaccination will reduce it to 0.0005 . Historically, \(25 \%\) of people are found to be susceptible.
3. What is the EMV of this questionnaire? Martin decides not to answer the questionnaire. He also decides that there is more than just his EMV to be considered in deciding whether or not to have the vaccination. The vaccination itself is likely to have side effects, but catching the disease would be very unpleasant. Martin estimates that he would find the effects of the disease 1000 times more unpleasant than the effects of the vaccination.
4. Analyse which course of action would minimise the unpleasantness for Martin.