A deductible comparison of cost per-loss, cost per-payment event, loss elimination ratio under exponentially and lognormally distributed severities

The maximum amount of losses retained by the insured under deductible policy modifications is usually set as part of the terms of the policy conditions. The objective of this study are to (i) estimate mean losses of an insured risk by means of the operational behavior of density function with deductible modifications and (ii) compare the mean severities under exponentially and log-normally distributed arbitrary policy in a cost per loss and cost per payment contingencies. The results show that despite the established fact in literature that log-normal severity distribution has a thicker tail than the exponential distribution, its cost per loss payment is correspondingly lower than the corresponding values of the exponential mean loss. Computational evidence over the trend of the change in the loss eliminated in the domain for which deductible has been defined revealed that the cost per loss amount is less than the cost per payment amount in the two models. The method can be used to estimate aggregate claims as the deductible level increases for every scheme holder and such that the estimated claims could be compared with the hypothetical observed claims which can be arrived at by applying the hypothetical deductible value to the background losses.