Diminishing Deductibles

tracyguo8tracyguo8
in Bahnemann.Chapter6

Hi,

Could you please prove the cdf for diminishing deductible and how do we interpret the cdf functions?

Thank you!

Comments

  • adminadmin

    Hi,

    To prove the CDF for the diminishing deductible you should think about the various payout intervals and apply conditional probability to ensure summing to 1.

    I'll show the second case, where the payout, x, comes from a loss that resulted in the diminishing deductible being applied. A payout of size x means the original loss, y, was somewhere between d and D. To get from y to x we use x = D/(D-d) * y which inverts to y = x *(D-d)/D

    However F_X(y) encompasses the probability of having a loss below size d, in which case we would always pay out 0. So it's necessary to subtract F_X(d). Lastly, we condition the probability by dividing by 1 - F_X(d).

    Remember, the CDF function for a diminishing deductible is expressing the payout x in terms of the original cumulative distribution function F_X where there are no deductible bounds applied.

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