Nonconditional franchise influence on insurance tariff

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Nonconditional franchise influence on insurance tariff

STATEMENT OF THE PROBLEM

Franchise as a specifical feature is inherent only to insurance business. Franshise is determined during insurance contract conclusion and determines loss compensation level which is not compensated by insurer. Usually franchise level is defined in agreement of insured and insurer and indicated in insurance contract. The franchise using leads to insurance tariff lowering as a result the insured saves his money. However special attention should be given to franchise level estimation as well as investigation of franchise influence on loss distribution and insurance tariffs.

ANALYSIS OF RECENT RESEARCH AND PUBLICATIONS

Works of V. D. Basilevich, A. A. Gvosdenko, М. G. Kaminkina, S. S. Gsadets etc are devoted to investigation of insurance contract peculiarities. Here franchise is examined as one of the insurance contract condition and much attention is devoted to juridical and organisation aspects of franchise using as well as to peculiarities of insurance franchise varieties. Works of V. V. Pusanenko, N. Miroshnichenko, М. Puslis are devoted to practical aspects of franchise using.

In aforenamed articles practical and juridical aspects of conditional and nonconditional franchise using have been investigated as well as peculiarities of insurance compensation calculation (in presence of the franchise) and peculiarities of insurance contract conclusion. But to statistical investigations of franchise influence on loss distribution and tariff value were not given consideration except few works (for example work by V. B. Kutukov). But in this work the loss distribution differential function were not examined.

THE PURPOSE OF THE ARTICLE

Main purpose of the article is investigation of franchise influence on insurance loss distribution and net tariff in risk insurance.

THE MAIN MATERIAL RESEARCH

Franchise in property insurance is defined as a form of insured's participation in loss compensation [1]. Another form of such participation is retention limit. The franchise implies self-insurance (that means risk on insurer's responsibility) with insurance. By franchise insurer is released from insignificant loss payment. That allows to simplify the procedure of compensation receiving and reduce insurance net tariffs.

In practical activity they can distingwish such kinds of franchise [1]:

1. Conditional franchise releases from insignificant loss payment (which doesn't exceed franchise value). In opposite case loss is compensated completely.

2. Nonconditional franchise independently from circumstances is eliminated from insured' commitment and releases him from insignificant loss payment (which is equal franchise value) independently from loss value. In this case insurance compensation is always equal to difference between loss and nonconditional franchise.

So difference between conditional and nonconditional franchise consists in the following: nonconditional franchise is a part of insurance loss which isn't compensated by insurer and is eliminated from insurer' commitment. Conditional franchise means that in the case when insurance loss doesn't exceed franchise value insurance compensation isn't paid. But in the case when insurance loss exceeds franchise value insurance compensation is paid completely.

Upon insured's view conditional franchise is more advantageous because even when loss is more than franchise the insurance compensation isn't paid completely. Moreover the contract with conditional is franchise comparing with nonconditional is more expensive.

One may distingwish such franchise functions [2]:

1) Optimizing function. Franchise allows not to compensate losses that correspond to constant and insignificant object damages. Cost of those damages is equal to insurer's expenses of expert estimation.

2) Prevention function. Franchise disciplines the insured about thorough accomplishment of a insurance contract, about highest honesty principle observance by careful attitude to insurance object and by avoidment insignificant losses;

3) Compensation function. Franchise compensates part of possible future losses on insurance expert examination;

4) Protective function. Franchise protects from possible insured's swindle for illegal insurance compensation receipt on nonexisting insignificant loss payment.

So according to insurance contract with franchise insured consents to take insignificant loss upon oneself. In the presence of nonconditional franchise compensation value always less than insurance loss; that leads to insurance premium decreasing [3].

Let perform statistical investigations of franchise influence on loss distribution and net-tariff value and obtain the loss distribution differential function.

Let's examine loss distribution in property insurance and consider that all statistical data are grouped in order of increasing. Let's introduce designation - Х(%) - loss value. Considering ranged sample of previous insured's loss results (n - volume of the sample) is known that

*1 * X2 < X3 < ... < Xn(1)

Using result sample of the relative frequency interval row have been built. Number of intervals is reccommended from 6 to 15. For ranged sample the relative frequency has been calculated.

Let A_; be the interval length. On each і-th interval the rectangle is built, its square is equal to ^Wi ^and altitude is equal to h =. In result distribution gystogram have been obtained [4].

The problem is to obtain loss distribution density knowing sample data. For the concrete sample dsitribution function F(x) has been chosen. Adequacy examination using Pierson' criterium X^і [4] is performed.

It is known [4], that irrespective to the form of F(x) by n the distribution %² approaches to %² distribution with degree of freedom is equal to v.

Let a be significance degree. Using %² distribution tables it is possible to find value of

%²(y,a) and compare with calculated value % [4]. If %² <%²(v,a), that distribution law F(x) corresponds with sample and statistical data.

Let's investigate loss distribution, and define its statistical characteristics and investigate franchise influence on net-tariff value in property insurance [5]. Insurance loss statistical data have been grouped into interval statistical series for random variables. Gystograms of statistical series have been built and hypotesis about loss distribution differential function has been proposed.

Let's use 5% nonconditional franchise and show that all insurance accidents with loss level less that 5% don't realize. Insurance accident number distribution considering franchise is decrease on 5% [6]. Moreover total compensation demand number is diminuted comparing with non-franchise contract. Attached to this frequency of fixed value demand is diminishes in consequence of total demand number lowering. So loss distribution gystograms for 10% and 15% franchise levels have been built (fig. 1):

Fig. 1. Gystograms of insurance loss Franchise level, %: a) 0; b) 5%; c) 10%; d) 15%.

Comparing table and sample Pierson's criterium %² shows that for all franchise levels sample criterium values doesn't exceed table ones. So there is no grounds to decline the hypothesis about insurance loss value exponential distribution existing. In general insurance loss value distribution differential function may be presented in such way [4]:

^f (Z) = A- exp^(-A-q)(2)

where Z - distribution parameter.

Let's define differential functions for insurance loss value distribution and numerical characteristics (mathematical expectation and standard deviation) and distribution parameter

value Z for different franchise values (table 1):

Table 1 - Characteristics of loss value distribution

Calculation of net-tariff (for 95% confidence probability) was perfomed using formula [1]:

^Тн = У ¦ p + tp- У ¦ p ¦ --^,(3)

\ np

where y - unprofitableness of insurance sum (y = 0,5), p - insurance accidents probability, П - number of insurance contracts, p - confidence probability, tp - critical value of Gauss distribution.

Nonconditional franchise influence on net tariff value in property insurance is presented at fig. 2:

Calculations shows that franchise level increasing up to 20-25% don't lead to significant change of insurance net tariff and is inexpedient. On other hand such franchise level is not suitable for insurer. So the maximal adequate franchise level is 10-15%.

In the presence of nonconditional franchise net-tariff value in property insurance decreases. Howewer, we have such effect only for specific insurance loss value distribution. For other loss distribution types franchise influence may differ and have other number characteristics.

CONCLUSIONS

Franchise releases from nesessity of paying losses which doesn't exceed previously stipulated level. Insured completely takes loss less than franchise value upon oneself and expects insurance tariff and payment diminution.

Insurance loss considering nonconditional franchise characterized by exponential distribution. Statistical characteristics and distribution parameter have been determined.

Nonconditional franchise level increasing leads to growth of insurance loss mathematical expectation and to insurance tariff lowering.

Investigation of dependence between franchise and insurance tariff in property insurance allows to define an optimal level of nonconditional franchise as 10-15%. The further investigation will be devoted to obtaining analytic formula of nonconditional and conditional franchise estimation.

LITERATURE

1. Александрова М. М. Страхування : підруч. / М. М. Александрова. -- К. : ЦУЛ, 2002. -- 208 с.

2. Пузаненко В. В. Франшиза в страхуванні / В. В. Пузаненко // Митна справа. -- 2011. -- № 4(76), ч.2. -- С. 389--393.

3. Мирошниченко Н. Франшиза в страховании: занимательная арифметика [Электронный ресурс] / Н. Мирошниченко, М. Пусліс. - Режим доступу : forinsurer.com/public/10/02/04/4024.

4. Герасимович А. И. Математическая статистика / А. И. Герасимович. -- Мн. : Высшая школа, 1983. -- 279 с.

5. Кутуков В. Б. Основы финансовой и страховой математики / В. Б. Кутуков. -- М. : Дело, 1998. -- 301 с.

6. Кисилева И. Ю. Влияние франшизы на распределение страхового убытка / И. Ю. Кисилева // Вісник Одеського національного університету. -- 2013. -- Т. 18, вип. 4. -- С. 51--53.

REFERENCES

1. Аleksandrova, М.М. (2002), Strakhuvannia [Insurance], textbook, TSUL, Kyiv, Ukraine.

2. Pusanenko, V.V. (2011), “Franchise in insurance”, Mytna sprava, no. 4(76), part 2, pp. 389-393.

3. Мiposhnichenko N. and Puslis, М. (2010), “Franchise in insurance: entertaining arifmetics”, available at: forinsurer.com/public/10/02/04/4024 (access September, 2013).

4. Gerasimovich, А.І. (1983), Matematicheskaya statistika [Mathematical statistics], Vysshaya shkola, Minsk, Belarus.

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