= Incremental second-claim threshold at the highest discount level
{title2=$L_1>c(\beta-\alpha),\qquad L_2>c\beta$}
For a <no claims discount system> with discounts $0,\alpha,\beta$, assume a policyholder starts at $\beta$, the first claim lowers next year's discount to $\alpha$, and a second claim in the same year lowers it to zero. With no further losses and undiscounted future premiums, claiming for the first loss costs $c(\beta-\alpha)$ in extra premium. Once that claim is submitted, the second claim adds $c\alpha$ next year and $c(\beta-\alpha)$ the year after, a total $c\beta$. Thus the second decision uses the incremental threshold, not the cost of both claims relative to no claim. For independent unit-rate <exponential distribution> losses the probabilities are $e^{-c(\beta-\alpha)}$ and $e^{-c\beta}$, respectively; the latter remains the conditional probability given submission of the first claim.
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