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Actuarial comparative analysis of natural premium

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Actuarial comparative analysis of natural premium

  1. 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. ABSTRACTInsurance contract is a legal contract and because of the legality,premium is one of the basic consideration for the acceptance of theinsurance risk Canning Vs Farquahar (1868) stated “NO PREMIUMNO INSURANCE” and Ivamy (1979) defined insurance as a contract.Based on these, it is necessary to know how companies determinetheir premium charges either on natural method and level premiummethod and it was found that premium under level premium wasbetter than natural premium and illustrations were made on thepossibilities and recommended that companies should use levelpremium method rather than natural premium.
  2. 2. 2KEYWORDSPremium, level premium, natural premium, actuarial valuation,surrender value, paid up policy. INTRODUCTIONThe 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 insuredpays to the insurance company in view of the risk inured, so that if aloss occur, the insurer will put the insured in the same financialposition he or she was prior to the loss (Ivamy: 1979) somecompanies charge level premium, while others charge naturalpremium.
  3. 3. 3HISTORY OF NATURAL PREMIUM.Natural premium is the situation in life policy where by the premiumcharge at the commencement of the contract continues to increaseas the age increases using a mortality table.Even a cursory glance at a modern mortality table will reveal that thechances of dying during any particular year varies remarkablyaccording to age.Thus, to take an example, a man who is aged 25 will pay lowerpremium, but the premium he is going to pay is higher as the ageincreases. The premium must steadily increase as the age rises,because the risk of death steadily increases and it must be ensuredthat each year’s claims are covered by each year’s premium. Theincrease would be sharp until the time when the premium wouldbecome prohibitive.The position might be modified if it were possible each year to securea large influx of younger lives, but in practice this has never beenfound to be the case.
  4. 4. 4The second difficulty is due to selection; this is the identification oflives, which from the point of view of mortality are inferior.There are two types of methods used to achieve this; one is byimposing a medical test each year on the participants or theimposition of the subsequent state of health ignored.If however, selection is made only at the time of original entry andthere is no medical test each year, the tendency would naturally befor more of the best and fittest lives than of the inferior lives toabandon 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 leadto more frequent deaths and premiums would still be furtherincreased in order to cover the claims.This therefore has made the natural premium system unworkable andthe system almost be completely abandoned. Many attempts hasbeen made to revive the scheme or even restrategise it, but all to noavail. This threaten to the development of an entirely different system,which is the level premium system.
  5. 5. 5The level premium system, is a system of premium calculation thatstipulates that a single percentage be collected uniformly throughoutthe duration of the policy, This system emphasizes that, if therefore alevel premium be charged throughout the duration of the policyduring a time of increasing risk, a premium will be payable during theearly years that is higher than is needed to meet the cost of the risk ofa claim. This is in order that there may be something in hand to meetthe cost of the greater risk in later years when the premium will beless than is required to cover the risk.HOW THE LEVEL PREMIUM SYSTEM WORKSThis is going to be illustrated on the assumptions that the group ofwhole life assurance is in a closed fund (with no new entrants oncethe 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 lifeassurance.
  6. 6. 6- The necessary knowledge is available which will enablepremiums 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 ofthe plans, and- Any margin for safety can be ignored.In the first year, there will be few deaths causing a moderateabsorption of the premiums; the balance – a very large one – will goto the reserve.There are no new entrants because it is a close fund, so that in thesecond year there will be slightly fewer premiums because of the factthat no premium would be collected from those who died in the firstyear. The claims will be slightly greater. The difference between thepremiums and the claims will again go to reserve.Each year the premium income will be slightly less and the claims willbe slightly more, with the balance still going to the reserves. The
  7. 7. 7reserve then gradually grows until comes a time when the premiumsbalance the claims and there will be nothing for reserve.The next year’s claims will slightly exceed premium and thedifference must be drawn from reserve. This reserve then graduallyreduces 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 enoughto pay the claim.This will be so where the assumptions as to interest, mortality andexpenses are exactly those experienced throughout the whole of theoperation.FEATURES OF THE LEVEL PREMIUM SYSTEM
  8. 8. 8The following is a summary of the features of the level premiumsystem:- 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. 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 THENATURAL AND LEVEL PREMIUM SYSTEMSReserves: consider whole life insurance policy of N1,000 issued toan individual aged 22. In the table below the net annual premium forthis policy is compared with the natural premiums at various aged ofthe insured. Net Annual Premium NaturalAge At Age 22 Premium22 13.28 2.53
  10. 10. 1023 13.28 2.6140 13.28 4.0351 13.28 12.9552 13.28 13.9575 13.28 86.4785 13.28 189.38In the illustration, it is seen that during the early years of the policythe insured is paying the company more than the year. By – year costof 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 premiumpayment offer the cost of insurance is placed by the company in areserve fund which earns interest at the same rate as that used incomputing the premium. At age 52, the cost of one year of insurancefor the first time exceeds the premium payment. Beginning then atage 52 and continuing each year there after so long as the policy is ineffect, the company withdraws from the reserve fund, sufficient tomake 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. 11throughout the life of the policy. In accordance with the CSO tableused here the reserve at age 99 would be 1000 v = $975.61 that isthe net single premium for a whole life assurance policy of N1000 atage 99.The reserve fund at the end of the year is called the “terminalreserve” for the policy year. The terminal reserve less a nominalcharge for expenses is called the “cash surrender value “ of thepolicy. The insured may borrow at any time the cash surrender valueof his policy without further collateral and the terminal reservebelongs to the insured as long as the policy is in force. He could aswell allow his policy lapse and either take the cash surrender value oruse it to purchase another insurance policy.MATHEMATICAL ILLUSTRATION FOR LEVEL PREMIUMPRACTICE.rv + px ax+zrv = Ax + r – Px ax+r= mx+r – mx . Nx+r
  12. 12. 12 Dx+r Nx Dx+rEXAMPLE: thFind the terminal reserve at the end of the 10 policy year for anordinary whole insurance policy of N1000 issued to an individualaged 22.100010 V = 1000 A32 - 13.28 a32= 100 m32 – 13.28 N3232 321000 m32 – 13.28 N32 D 32 = 50,165,505 416,507 = N120.44From the above, the following conclusions and recommendations willbe made.
  13. 13. 13 CONCLUSIONS1. 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. RECOMMENDATIONSFrom this work, the researcher recommended that level premium isbetter than natural premium and recommended that policy holdersand insurance companies should consider level premiums the bestoption to natural premium REFERENCESAyres F. (1983): Mathematics of Finance, Aslan Student Edition.Marshal C. (1989): Insurance of the Person Chartered InsuranceInstitute London.
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