Bonuses, Dividends, and Refunds

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 L.  No refund is paid if losses exceed $100.  The refund amount R can be expressed as

R = 0.15 \max (0, 100-L)

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

\begin{array}{rll} R &=& 0.15 \max (100-100,100-L) \\ &=& 0.15(100-\min (100,L)) \end{array}

So the expected value is given by

E[R] = 0.15(100 - E[L \wedge 100])

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