If a policy pays a reward to the participants when losses are below a certain level, this is a particular type of problem which Weishaus calls a “bonus” problem. The bonus, dividend, or refund amount is expressed as a maximum between 0 and the refunded amount. For example, a 15% refund is paid on the difference between the $100 premium and the loss . No refund is paid if losses exceed $100. The refund amount can be expressed as

The key to finding the expected refund is knowing how to manipulate the max function and rewrite it as a min. We can rewrite as

So the expected value is given by

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