Actuarial comparative analysis of natural premium
- 1. 1 ACTUARIAL COMPARATIVE ANALYSIS OF NATURAL PREMIUM AND LEVEL PREMIUM AND HOW LEVEL PREMIUM WORKS. BY NWITE SUNDAY C. A RESEARCH STUDENT AND LECTURER DEPARTMENT OF BANKING AND FINANCE. EBONYI STATE UNIVERSITY ABAKALIKI. ABSTRACT Insurance contract is a legal contract and because of the legality, premium is one of the basic consideration for the acceptance of the insurance risk Canning Vs Farquahar (1868) stated NO PREMIUM NO INSURANCE and Ivamy (1979) defined insurance as a contract. Based on these, it is necessary to know how companies determine their premium charges either on natural method and level premium method and it was found that premium under level premium was better than natural premium and illustrations were made on the possibilities and recommended that companies should use level premium method rather than natural premium.
- 2. 2 KEYWORDS Premium, level premium, natural premium, actuarial valuation, surrender value, paid up policy. INTRODUCTION The contract of insurance is a contract that is based on utmost good faith and before the contract becomes enforceable, there must be consideration. Consideration can therefore be defined as the premium the insured pays to the insurance company in view of the risk inured, so that if a loss occur, the insurer will put the insured in the same financial position he or she was prior to the loss (Ivamy: 1979) some companies charge level premium, while others charge natural premium.
- 3. 3 HISTORY OF NATURAL PREMIUM. Natural premium is the situation in life policy where by the premium charge at the commencement of the contract continues to increase as the age increases using a mortality table. Even a cursory glance at a modern mortality table will reveal that the chances of dying during any particular year varies remarkably according to age. Thus, to take an example, a man who is aged 25 will pay lower premium, but the premium he is going to pay is higher as the age increases. The premium must steadily increase as the age rises, because the risk of death steadily increases and it must be ensured that each years claims are covered by each years premium. The increase would be sharp until the time when the premium would become prohibitive. The position might be modified if it were possible each year to secure a large influx of younger lives, but in practice this has never been found to be the case.
- 4. 4 The second difficulty is due to selection; this is the identification of lives, which from the point of view of mortality are inferior. There are two types of methods used to achieve this; one is by imposing a medical test each year on the participants or the imposition of the subsequent state of health ignored. If however, selection is made only at the time of original entry and there is no medical test each year, the tendency would naturally be for more of the best and fittest lives than of the inferior lives to abandon the scheme when the premium begin to rise sharply, occurring to the greater chance of death caused by increasing age. In this case more of the inferior lives would be left which would lead to more frequent deaths and premiums would still be further increased in order to cover the claims. This therefore has made the natural premium system unworkable and the system almost be completely abandoned. Many attempts has been made to revive the scheme or even restrategise it, but all to no avail. This threaten to the development of an entirely different system, which is the level premium system.
- 5. 5 The level premium system, is a system of premium calculation that stipulates that a single percentage be collected uniformly throughout the duration of the policy, This system emphasizes that, if therefore a level premium be charged throughout the duration of the policy during a time of increasing risk, a premium will be payable during the early years that is higher than is needed to meet the cost of the risk of a claim. This is in order that there may be something in hand to meet the cost of the greater risk in later years when the premium will be less than is required to cover the risk. HOW THE LEVEL PREMIUM SYSTEM WORKS This is going to be illustrated on the assumptions that the group of whole life assurance is in a closed fund (with no new entrants once the scheme has started) it may also be assumed that: - There is a large body of new entrants of a given age (say,25) all of whom have been selected by medical examination for life assurance.
- 6. 6 - The necessary knowledge is available which will enable premiums be calculated scientifically. - The expenses of ruining the scheme can be ignored. - Each policy remains in force until the death of the life assured, that is none of the policies is surrendered or made paid-up. - No other circumstances arise which cause any modification of the plans, and - Any margin for safety can be ignored. In the first year, there will be few deaths causing a moderate absorption of the premiums; the balance a very large one will go to the reserve. There are no new entrants because it is a close fund, so that in the second year there will be slightly fewer premiums because of the fact that no premium would be collected from those who died in the first year. The claims will be slightly greater. The difference between the premiums and the claims will again go to reserve. Each year the premium income will be slightly less and the claims will be slightly more, with the balance still going to the reserves. The
- 7. 7 reserve then gradually grows until comes a time when the premiums balance the claims and there will be nothing for reserve. The next years claims will slightly exceed premium and the difference must be drawn from reserve. This reserve then gradually reduces with every year because more claim will exceed premiums, until finally when one life is left in. He pays his last premium and dies, and last premium with the residue of the reserve is sufficient enough to pay the claim. This will be so where the assumptions as to interest, mortality and expenses are exactly those experienced throughout the whole of the operation. FEATURES OF THE LEVEL PREMIUM SYSTEM
- 8. 8 The following is a summary of the features of the level premium system: - The total reserve in a closed group of lives (that is, with no new entrants) increases to a maximum and then decreases. - The reserves for any one particular policy steadily increases throughout its duration steeply at first and more gradually later on. - The policy period is treated as a whole. Once the premium is fixed it cannot be altered. - The premium must therefore be scientifically fixed. Knowledge of the probable course of mortality is required, hence the investigations into the mortality of the past and the production of mortality tables. - Reserves will be invested at interest, so that knowledge of compound interest is required. - Allowance must be made for expenses of management, commission and a margin for adverse features.
- 9. 9 - Also to be noted in the assessment of premium to be charged, it is necessary, therefore, to take into account not only the chance of death at any particular age but also, - The rate of interest which can be earned on reserve if invested and; - The additional amount (called loading) which must be added to the premium to cover expenses and to provide a reasonable safety margin. THE ACTUARIAL COMPARATIVE ANALYSIS OF THE NATURAL AND LEVEL PREMIUM SYSTEMS Reserves: consider whole life insurance policy of N1,000 issued to an individual aged 22. In the table below the net annual premium for this policy is compared with the natural premiums at various aged of the insured. Net Annual Premium Natural Age At Age 22 Premium 22 13.28 2.53
- 10. 10 23 13.28 2.61 40 13.28 4.03 51 13.28 12.95 52 13.28 13.95 75 13.28 86.47 85 13.28 189.38 In the illustration, it is seen that during the early years of the policy the insured is paying the company more than the year. By year cost of the insurance, 13.28 2.53 = $10.75 in the first years, and 13,28 2.61 =$10.67 the second year. Each excess of annual premium payment offer the cost of insurance is placed by the company in a reserve fund which earns interest at the same rate as that used in computing the premium. At age 52, the cost of one year of insurance for the first time exceeds the premium payment. Beginning then at age 52 and continuing each year there after so long as the policy is in effect, the company withdraws from the reserve fund, sufficient to make up the difference 13.95 13.28 = $0.69 at age 52 and 86.47 13.28 = N73.19 at age 75. The reserve fund on this policy increases
- 11. 11 throughout the life of the policy. In accordance with the CSO table used here the reserve at age 99 would be 1000 v = $975.61 that is the net single premium for a whole life assurance policy of N1000 at age 99. The reserve fund at the end of the year is called the terminal reserve for the policy year. The terminal reserve less a nominal charge for expenses is called the cash surrender value of the policy. The insured may borrow at any time the cash surrender value of his policy without further collateral and the terminal reserve belongs to the insured as long as the policy is in force. He could as well allow his policy lapse and either take the cash surrender value or use it to purchase another insurance policy. MATHEMATICAL ILLUSTRATION FOR LEVEL PREMIUM PRACTICE. rv + px ax+z rv = Ax + r Px ax+r = mx+r mx . Nx+r
- 12. 12 Dx+r Nx Dx+r EXAMPLE: th Find the terminal reserve at the end of the 10 policy year for an ordinary whole insurance policy of N1000 issued to an individual aged 22. 100010 V = 1000 A32 - 13.28 a32 = 100 m32 13.28 N32 32 32 1000 m32 13.28 N32 D 32 = 50,165,505 416,507 = N120.44 From the above, the following conclusions and recommendations will be made.
- 13. 13 CONCLUSIONS 1. Natural premium considers the risk yearly. 2. Situation of the risk may change the policy. 3. The premium increases as the age increases in natural premium. 4. Level premium is the best where the same premium is paid. RECOMMENDATIONS From this work, the researcher recommended that level premium is better than natural premium and recommended that policy holders and insurance companies should consider level premiums the best option to natural premium REFERENCES Ayres F. (1983): Mathematics of Finance, Aslan Student Edition. Marshal C. (1989): Insurance of the Person Chartered Insurance Institute London.
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