The insurance premium and the demand for insurance from the perspective of portfolio theory

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1 The insurance premium and the demand for insurance from the perspective of portfolio theory Adriana Elena Simion (Istrate The Bucharest Academy of Economic Studies E- mail: Abstract: An analysis of underwritten premiums in the first months of 9 reported the same period of 8 reflects a decrease in interest from customers for insurance.the main reason is the reduce of cashflow in the customers budget, and thus, the amounts allocated to the insurance contracts, as a consequence of the financial crisis facing our country. Even if in the insurance area, the financial crisis not yet felt strong, during the first months of this year was a slight decrease in demand for voluntary policies such as providing CASCO insurance or home insurance. Decrease premiums for insurance contracts, the property type can be induced by the rumors that such policy could become mandatory from 1 July 9. Companies, taken individually, are affected by how they treat customers. A company in which the restructuring, lose customers because they can go with the employee. Studying the value of the insurance premium is very important because it directly influences the demand, and this in turn affects the structure of portfolio insurance. In this study aims to determine the solution to maintain the number of customers insurance companies. This paper presents some of the results obtained by authors, in the grant CERCETARI EXPLORATORII PRIVIND ELABORAREA UNUI SISTEM INTELIGENT DE OPTIMIZARE A DECIZIILOR FINANCIARE, number 185 of the National Plan for Research, Development and Innovation PN II - Program Ideas, director professor PhD Adrian Victor BADESCU. Keywords: insurance premium, insurance demand, portfolio JEL Classification: D8- Microeconomics, Information, Knowledge, and Uncertainty- General According to data provided to the Commission of Insurance Supervisors, by the insurers, the companies recorded a decrease of only 4% during the first two months of this year, estimate that the 9 will end with an increase of 5% -6% of the insurance market as a whole. The volume of underwritten premiums fell from 1.69 billion lei in first two months of 8, to 1.6 billion lei, with considerable decrease of 4% for general insurance and 5% for life insurance. The decline in non-life insurance is due to reduce by 6% of the amount of premiums assurance type of mandatory auto liability and 1% for insurance type Casco. Falling auto policy is a consequence of the crisis that is in the lease sector, and the credit crunch. Earlier this year, leaders of the insurance companies estimated a nominal growth of insurance market in Romania more optimistic, about 1%. After years of mergers, they decided that 9 is a year dedicated customers, the main objective being to preserve the existing, but wishes and attracting new customers. For a more detailed study is required of certain terms such as explicit rate pricing, coverage of insurance, the amount secured, insured and first period of insurance. Tariff quotas are set in percentages, and are applied to the assured sum, depending on certain clauses apply to increases or reductions. Coverage of insurance is represented by the conditions of insurance. The sum insured can be an asset according to the invoice value, but in the absence of that, the market value of the property at the time of insuracnce contract. The insurace period is the time period provided that the subject is covered by an insurance contract. 316

2 The insurance premium is determined by insurance companies based on an analysis of coverage of insurance, the sum assured, term insurance, the nature of the object insured damage earlier statistics and market situation. The premium is paying for a regular insurance policy. In fact, the insurance premium is the amount of money charged by insurance companies to cover a risk, and may vary for a product according to the insurance company providing the insurance policy. Insurance agents calculated first estimated using information provided by the customer base for more companies. This may require the company to choose the lowest price, but must be cautious, because in most cases and the level of coverage may be lower than in other situations. The premium is determined on the basis of certain statistics, not necessarily taking into account individual habits. For example, a potential customer aged years, owner of a sports car, pay a more expensive premium for insurance than a woman of 45 years who drives a Sedan. Even if both prospects are not recorded incidents, the insurance company believes that a young driver with a faster car has a higher risk for accidents. Therefore, the insurance premium will be calculated differently, sensitive. In general, a more expensive car or faster will have a higher premium of insurance, because the invoice price is higher and the risk assumed by the company is greater. The same philosophy applies in the case of medical insurance and health. For example, statistically non-smokers live a healthier life than smokers, or construction workers can have serious accidents at work compared with economists. A worker aged 55 years, who smoke, pay an insurance premium of more than an economist for 3 years. In general, the insurance premium is paid regularly: monthly, quarterly or half-yearly, at regular intervals. Where the holder of the insurance fails to pay a rate to a certain grace period, the insurance policy will terminate. The value rates may be fixed or may vary. For example, if a policy CASCO, after a claim of the insureds fault, the insurance premium will increase by introducing reunification sum insured. Currently, most states have civil liability insurance auto compulsory. Payment of insurance premium can be regarded as a waste of money, but it brings a psychological comfort. In general, the premium varies from one company to another. Insurance brokers play an important role in finding suitable product for the customer, through assessments of each company depending on the potential customer. The best offer for the customer is considered to be the one with the lowest rate. Also, insurance premiums may be calculated by the potential customers on specialized websites, and the conditions compared. In general, insurance premium increases with increasing perception of risk assumed by the insurance company. For the medical insurance, insurance premium for smokers is higher than for non-smokers, because the insurance company deemed the risk of a smoker is higher than for a nonsmoker. Therefore, the insurance premium is directly proportional to the associated risks. In the case of life insurance, the companies consider that a senior is more prone to death than a younger. Thus, the premium paid by the older person is greater than that paid by the young person, because under normal circumstances a young person has the life expectancy greater. Study on the premium of insurance is very important because it directly influences the demand and this in turn affects the structure of portfolio insurance. In what follows I present a model of demand for insurance. This model takes into account the unique risk, using the fact that the decision of insurance may be taken without regard to take risks other exogenous or endogenous. Exogenous risks include uncontrollable risks of individual assets, for example, the gains of globalization on financial markets. 317

3 Endogenous nature of the risks include the risks arising from information asymmetry, such as moral hazard and adverse selection. An individual interested in a policy has an insurance amount W and may record a loss of a damage L, with the value of probability π. Thus, can be define the lottery ( W L, W. If it completes an insurance policy, the customer must pay a premium P pc, which p represents the rate of insurance premium and C is the sum insured, the level of coverage. Thus, the lottery shall be amended as follows: ( W pc L + C, W pc (1 In case of total damage, the damage is equal to the sum assured, then C L. In this case, the amount of money available is W pc, whether damage occurs or not, so the policyholder would prefer to lose the insurance premium, than to bear a loss uncertain L. It is assumed that this utility is the von Neumann-Morgenstern U (W individual utility function. Thus, the function U (W is continuous and twice derivatives; marginal utility is positive and decreasing in relation to the amount of money available to the individual. Considering these assumptions, the insurance contract will be signed if and only if the sum insured, in the marginal utility of insurance is higher than expected utility if uninsured. In this situation: πu ( W pc L + C + π U ( W pc > πu ( W L + (1 π U ( W. ( The decision of optimal insurance are analyzed by solving the problem: max E U W π U W pc L + C + π U W pc. (3 C ( ( ( ( ( 1 In order to maximize expected utility, should be solved the first order condition: π p U ( W pc L + C p π U ( W pc. (4 Second order condition is: d E( U ( W π p U ( W pc L + C + π p U ( W pc < dc The following applies Bernoullis principle, which states that an individual at risk front will sign a contract of total insurance, when the premium is fair. Assuming that the premium is fair, that is p π, and substituting π with p in the equation (4 obtain the following equation: U ( W pc L + C U ( W pc (5 W pc L + C W pc. Therefore, if the insurance premium is correctly set, the conclusion of a insurance contract is optimal. Derived demand equation in that case of a logarithmic utility, provided by the first order equation (4 implies: π p p π (6 W L + p C W pc Solving this equation to obtain the amount C provided in the form: ( π pl + ( p π W C. (7 p( p The effect of changing the initial amount available to the insured. The effect of changing the initial amount available on the demand for insurance by considering the optimal differentiation of the sum insured C, given the equation (7, in relation to W : C p π. (8 W p(1 p 318

4 If the rate of insurance premium equal the probability of occurrence of damage, p π, when changing the initial amount held by an individual does not affect the demand for insurance, since insurance is fully optimal, regardless of value W. If insurance premium is set incorrectly, for example, if p > π, when demand for insurance is inversely proportional to the initial amount available. The effect of changing the damage occurence probability. To analyze this effect is to analyze the relationship between the optimum coverage of the damage and the probability of its occurrence, by differentiating the sum insured in relation to π : C Lp W π p p. (9 Since one can not spend more than the amount available, this report numerator must be negative, as denominator. This implies that the demand for insurance is higher, the greater the possibility of occurrence of damage. Consequently, there is a positive relationship between the optimal level of coverage and probability of occurrence of damage. The effect of changing gravity damage. By differentiating the sum insured against the damage, to obtain the best insurance in the event of changes on the damage severity: C π 1. (1 L p 1 The demand for insurance is definitely related to the severity of damage. The change rate of insurance premium. The relation between the optimal level of cover and premium for insurance shall be obtained by differentiating the sum insured in relation to the probability of occurrence of damage: C Lp ( π + ( p + π pπ W π ( Lp + W + p ( W L pπw. (11 p ( p p ( p p In general, we expect demand for insurance is inversely proportional to the insurance premium. In equation (11, denominator is obviously positive. π Lp + W is clearly positive, for the second term For the numerator, the first term ( ( ( W L for the third term ( pπw p (the equality being valid when full amount initially available is the risk, and is obviously negative. Therefore, demand for insurance is related to the insurance premium is positive when p W > π ( Lp + W + p ( W L π. A positive relationship between the sum insured C and the probability of occurrence of damage p suggests that insurance is an "inferior" to the effect of income is greater than the substitution effect, so that when the first insurance decreases and the demand for insurance decreases. In equation (11, ( Lp W + p ( W L pπw may be interpreted as the effect of income and π + the substitution effect. However, Hoy and Robson (1981 showed that insurance can not be regarded as less good, if the coefficient of relative risk aversion is less than or equal to one. Since the coefficient of relative risk aversion for the logarithmic utility function is equal to one, this implies that the sign of equation (11 to be negative, for example, demand for insurance is inversely proportional to the insurance premium. The Optimal amount deductible. For this analysis is based on Arrows theorem: If an insurance company is willing to offer an insurance policy to cover loss that is an interested buyer at a premium which depends on actuarial value of the premium insurance then the policy chosen by a policyholder front will be at risk for a coverage of 1% above the minimum deductible. In general, the premium will be greater than the actuarial value, is just two policies with the same actuarial value to be offered by the same company at the same insurance premium. 319

5 An alternative use of Arrow theorem is optimality of reinsurance of Gerbers Pafumi. Consider the following insurance policies: - a first policy with a deductible premium, which paid the damage in case of occurrence of the risky I ( L, d Max[ L d,] - a policy, for which the damage amount I (L is paid, with I ( L L. In other words, this bill inculde the deductibility, but may present specific characteristics and other types of contracts, as co-insurance. As shown by the Arrow, it is assumed that these policies have the same actuarial value and signed for the same insurance premium. Deductible policy will be preferred general policy, if the following condition is fulfilled: E( U( W L + I ( L E( U ( W L + Max( L d,. (1 Using observations of Gerber and Pafumi, the domonstation starts from the simple observation that the concave curve is located below the tangent, for example: U ( y U ( x + U ( x( y x. (13 And substituting y W L + I ( L and x W L + max ( L d, in the relationship (13 is obtained: U W L + I( L U W L + max L d, + U x I L max L,. (14 ( ( ( ( ( ( ( d Since both policies have the same actuarial value then ( I( L max ( L d, E. Therefore, when calculating the expected value for both sides of equation (14, was obtained Arrow theorem. In conclusion, the deductible premium must be different from zero if the premium includes a proportional loading factor different from zero. For this analysis is assumed that a person has available an initial amount W present likely to suffer damage L. Where is the P 1 + λ π L d, which d is the insurance policy deductible is paid an insurance premium ( ( amount of deductible and λ is charging premium. Thus, the uninsured represented by lottery( W L, W may change in lottery ensured ( W P d, W P. Therefore, the decision of insurance requires solving equation: max E( U ( W πu ( W π ( L d d + π U ( W π ( L d. (15 d First order condition is: π π U ( W π ( L d π πu ( W π ( L d d. (16 Assuming λ, then relation (16 turns into: U ( W π ( L d U ( W π ( L d d. (17 If λ, then d. However, if λ >, then d must be greater than, otherwise, the marginal utility of the amount available in the occurrence of unforeseen damage and in situations where damages are not, can not be equal. Assuming that the utility is logarithmic form, then the first order condition is given by the relationship: π π π π. (18 W π ( L d W π ( L d d For this equation to obtain the value of d : λ( π π L W d. (19 ( π Therefore, deductibility depends on the loading of premium for insurance p, the occurrence probability of damage λ, the amount of available initially W, and damage severity L. For this case the relations are valid: 3

6 ( π π ( L W ( π W + >, optimal deductibility is directly related to λ loading the insurance premium λ( W L >, optimal deductibility is directly linked to the probability π ( π of occurrence of damage λ >, optimal deductibility is directly related to the initial W π value of the amount available πλ <, optimal deductibility is inversely proportional to the degree of L ( π severity of damage. Exemplification. An individual has the amount of 75 eur, and assessing the risks to which is subject to the dwelling, it is estimated that the damage value will be 5 eur, with a probability of 5%. Considering that the probability of registering damage equal share of policy practiced by the insurance company, then the amount that the company may be a cover: C ( π pl + ( p π p( p 1 W (,5,5,5 (, It is normal to have a cover damage entirely, because according to Bernoullis principle, the damage is covered entirely by the sum insured when the premium rate offered by the insurance company is equal to the probability of occurrence of damage. The insurance company applies a loading rate of % premium. In these circumstances the new premium rate will be: p 1 p + p λ,5 +,5,6. The sum insured will change so: ( π p L + ( p1 π p ( p (,5,6 5 + (,6,5,6 (,6 1 W C 31,5 1 1,4 The coverage decreases from 5 eur to 31.5 eur, because the insurance became more expensive. The customer is willing to sign a bill to increase the coverage limit as part of premium winning utility from buying an insurance policy. Where the amount available for insurance increases from 75 eur eur in 1, then: (,5,6 5 + (,6,5,6 (, C 8,33.,4 The coverage decreases with increasing initial amount available, because the marginal utility decrease. Practical utility of the loss limit cover is less problematic for a rich person than for a poor person. Considering that its value increases from 5 eur to 6 eur, then: (,5,6 6 + (,6,5,6 (, C 333,33.,4 In this situation the level of coverage increases because of increased severity of damage, loss involves greater individual gains from transfer of risk to an insurer, even if the insurance premium is fairly established. As regards the deductibility if the amount available is 75 eur, 5 eur loss, the probability of occurrence of damage is.5 and factor loading of the premium rate is., then: 31

7 λ( π π L W ( π (,5 1, (,5 1,, 75 d 156,5. 1,,48 Optimal deductibility is directly proportional to the factor încăracre share premium. If this factor increases to.4, then: λ( π π L W,4 (,5 1, d 357,14. ( π 1,4 (,5 1,4,4 Also in this deductibility is directly linked to the likelihood of injury and the initial amount available. REFERENCES: [1] Ben-Haim Yakov, Decisions under Severe Uncertainty, Info Gap, 6 [] Cairns A. J. G., A Discussion of Parameter and Model Uncertainty in Insurance, Insurance: Mathematics and Economics, number 7, p , [3] Dobre I., Bădescu A.V., Păuna L., Teoria Deciziei, ASE, Bucureşti, 7 [4] Jatinder N.D. Gupta, Manuel Mora T., Intelligent Decision-making Support Systems, Springer, p. 5-45, 7 [5] Wang, S. S., A Universal Framework for Pricing Financial and Insurance Risks, Astin Bulletin, number 3, p , 3