# Franchise and its types - Actuarial calculations

## Franchise and its types

A franchise (L ) is a non-refundable loss. Agreements with the franchise leave insignificant losses on the retention of the policyholder in exchange for a reduction in the insurance premium. Thus, the insurance company frees itself from small losses, the registration and registration of which often requires the same expenses as large ones, and the insured has the opportunity to save on the insurance premium and at the same time protect himself against serious serious damages.

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The franchise is especially often used in motor insurance, as it is characterized by a large number of fairly small damages.

There are three types of franchise.

The unconditional (deductible) deductible. The damage is compensated to the insured if the loss exceeds the established amount - the franchise L, minus the franchise loss see Figure 1.9). If the loss is less than L, then no refund is made:

(1.5)

Thus, (1.6)

Agreements with an unconditional franchise (in the English literature it is called deductible, simple deduction) are the most common in practice.

Conditional (unreadable) deductible. If the damage exceeds L, it is paid in full. Losses that do not exceed the value of L, are not reimbursed (Figure 1.10):

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(1.7)

Aggregate (mixed) deductibles. All losses incurred by the insured are added over a certain period of time, and the deductible amount L:

(1.8)

The aggregate franchise will usually give payout results that are intermediate between the unconditional and conditional franchise, and the least common.

EXAMPLE 1.5

The policyholder has an object worth C = 1 million cu and concludes an insurance contract with a franchise L = 150 thousand cu. What kind of payments will the insurance company provide to him, if in a year there come insured events that caused damage in the amount of X, = 100 thousand cu. and X 2 , = 800 thousand cu, if the franchise under the terms of the contract is: a) unconditional; b) conditional; c) aggregate?

Solution

Under the contract with both franchises (a and b) after the first insured event, the insured should not even apply to the insurance company, since the first damage equal to 100 thousand y ., does not exceed the deductible and is not refundable.

Under a contract with an unconditional (deductible) deductible, the second damages exceeding the deductible will be partially paid, minus the franchise. 150,000 cu. Thus, the payments of the insurance company will be: Y] = 0; Y 2 = X 2 - 1 = 800 - 150 = 650 thousand cu

Under the contract with a conditional (unreadable) franchise, the second damage exceeding the deductible will be paid in full. Thus, the payments of the insurance company will be: Y = 0: Y 2 = X 2 < strong> = = 800 thousand cu

Under the agreement with the cumulative franchise, all losses that occurred during the year are added together, and the deductible deducts the deductible. Thus, the payment of the insurance company to the insured will be (1.8):

Comparing the results obtained in examples 1.2-1.5, we get the following total payments to the insured for all six treaties examined:

 No. Agreement Amount Payouts 1 Full Protection 100 800 900 2 Proportional protection, 5 = 700 70 560 630 3 Protection by the rule of the 1st risk, 5 = 700 100 700 800 4 Unconditional franchise, L = 150 0 650 650 5 Conditional franchise, L = 150 0 800 800 6 Aggregate franchise, L = 150 750
As can be seen from the obtained results, the largest payments after the full protection contract are provided by contracts under the first risk rule and with a conditional franchise, since they provide not only partial but also full payments for a significant part of the damages, limiting the payments of the insurer from above and from below, respectively; the smallest payments - under the contract of proportional protection and with an unconditional franchise, since they require the participation of the insured in all payments.

As noted earlier and seen from the results obtained in Examples 1.2-1.5, for an insurance company, the application of partial (proportional and disproportionate) insurance contracts means a reduction of its liability, while part of the risk remains on the insured. Therefore, the policyholder has the right to expect a corresponding decrease in the insurance premium.

The discount from the tariff should correspond to the expected reduction in payment under the contract, i.e. reduction of risk expression. Since the damage under the contract is not necessarily evenly distributed, it is unreasonable to provide a discount proportionally to the share, for example, of the deductible in the insured amount. It is necessary to calculate (estimate) the parameters of the new distribution - reduced in case of a conditional franchise or the 1st risk rule, and reduced and reduced by the amount of the deductible in the case of an unconditional franchise.

In addition, when using a franchise, since for insured events, the loss in which did not exceed L, the insurer does not pay compensation, the probability of payment on the insured event will decrease. Also, the distribution of the number of payments will change.

EXAMPLE 1.6

Insurance company insures cars worth 15 thousand cu. 1 year unconditional (deductible) franchise (deductible) 1 th. USD During the validity of the insurance policy in the course of the year with a probability of 0.04 can be applied to the vehicle partially damaged and with probability 0.02 possible total loss. In case of partial damage, the damage X (in thousand cu) is distributed continuously with the density of the form:

What is the expectation of payments?

a) 320 cu: b) 328 cu; c) 352 cu; d) 380 cu; e) 540 cu

Solution

Y - payments of the insurance company in this case have a mixed distribution - discrete-continuous. With a probability of 0.04 for partial damage, payments have a continuous character of the form (1.5):

as well as possible discrete payments - with a probability of 0.02: F = 15 - 1 = 14 thousand dollars; with a probability of 0.94 payments will not be at all, i.e. F = 0. Thus, the payments F in thousands of US dollars are equal:

The mathematical expectation of a mixed random variable is found as the sum of the mathematical expectations of the discrete and continuous components:

Thus, M (Y) = 0.328 thousand cu = 328 cu

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