Chapter 10

Calculation of Premium

Slide prepared by: Abdullah Al Yousuf KhanAssistant Professor IUBAT 10

ChapterMcGraw-Hill/IrwinCopyright 2006 by The McGraw-Hill Companies, Inc. All rights reserved.Calculation of PremiumPremiums are two types;Net Premium, and Gross premiumBoth types are further sub-divided into two parts; Single Premium, and Level Premium2Basic Characteristics of PremiumsNet PremiumBased on; The mortality and Interest Gross PremiumBased on;The mortality rate Assumed interest rateExpenses, and Bonus

3Basic Characteristics of PremiumsSingle PremiumIs paid in lump sumLevel PremiumIs paid periodically in installmentsMay be yearly, half-yearly, quarterly, or monthly4Firstly, net single premium is calculated and other premiums are calculated based on this calculation.Net Single PremiumReceived by the insurer in a lump sum.Adequate Generates returnsTo be able to pay the claim (of agreed amount)At death, at maturity, or at surrenderDoes not provide for management expenses and contingency5Net Single Premium; Steps for CalculationDetermine; What constitutes a claim; (a) death, (b) survival, or (c) bothWhen claims are paid; (a) at the beginning, (b) at the end, (c) during the year.The number of insured.The duration of the policy.The probable number of claims per year.The value of claims per yearThe number of years of interest involved and find the present value.The PV of the claims for each year.The PV of all future claims.The net single premium (i.e. the PV of future claims) divided by assumed number of policy buyer.

6Assumptions underlying Rate ComputationsAs many policies of given type are being issued as are the number of persons.Premiums are collected in advance or in the beginning of the period.All collections are immediately invested and will remain invested until money is needed for the payment of claims.The insurer will receive an assumed rate of interest. 7Continues in next slide ...Assumptions underlying Rate ComputationsThe interest or dividend or any return of the invested funds is immediately reinvested.Mortality rate will be the same as given in the mortality table and will be uniformly distributed throughout the year.All policies are of the same amount, say e.g. tk. 1,000.Claims will be paid at the end of the period. 8Calculation of Net-Single PremiumTerm Insurance;Simplest type of contract.Payment is made only when the life assured dies within the term specified.Nothing will be paid if death does not occur during the designated term.Also called temporary insurance.Premium is received in advance and will not be returned if life assured survived.Premium is paid only once in a single sum at the inception of the policy.Claims will be paid at the end of the year, not at the end of the term. The probability of death in each year with the PV of the claim will be calculated Because death may occur at any time and the insurer may be required to pay.

9Term Insurance; Example of Net-Single Premium Calculation Assume that the period of term insurance is 5 years. Rate of return on investment is 3 percent. The person is proposing at the age of 40 for a period of 5 years. The mortality experience will be like shown in the Oriental 1953-54 Experience Life Table; 10Age Number of Persons LivingNumber of DeathsMortality Rate per 1,0004096,4632732.834196,1903023.144295,8883363.504395,5523753.924495,1774184.394594,7594764.03Table 10.1: Oriental 1953-54 Experience Life Table, Ultimate TableAssumptions for Calculation Each dead person will be paid tk. 1,000.Because the insurer will earn a fixed return on the investment, only PV of the claim should be taken as premium. Thus the net-single premium for each year will be calculated;

Number of Deaths Amount of Claims PV of tk.1 = PV of Claims11Continues in next slide ...Table 10.2: Net Single Premium for 5-Year Term Policy12Years of Insurance Age Number of Persons LivingNumber of DeathsAmount of Claims per DeathsPV of tk. 1PV ofClaim 4x5x6123456714096,4632731,0000.971265,08324196,1903021,0000.943284,78634295,8883361,0000.915307,44044395,5523751,0000.888333,00054495,1774181,0000.863360,734PV of all claims tk. 1,551,043Total premium to be charged on all 96,463 policies = tk. 1,551,043Therefore, premium per policy = tk. 1,551,043/ 96,463 = tk. 16.18Continues in next slide ...Calculation of Net Single PremiumThe PV of the 1st year will be calculated for one year in the 1st year of insurance because death of death occurs during the 1st year, the insurer can earn interest at least for one.Similarly, in the 2nd year of insurance, the PV of tk. 1 is calculated for two years at 3 percent per annum. The equation for calculating PV is = (who does not know this?) Where, P stands for Present ValueS stands for the amount of which PV is to be calculated i stands for the rate of interest, and n for number of years for which PV is to be calculated.13

Calculation of Net Single PremiumThe PV of claim for 1st year will be 273 x 1,000 x 0.971 = 265,083The total amount of claim is 273 x 1,000 = 273,000.The formula to calculate the Net Single Premium is;

Where,V represents the value of tk. 1dx represents deathslx represents number of living in the beginning

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Other method of calculation of Single Policy Premium by taking into account of probability of death15Years of Insurance Age Probability of deaths expressed in unitsAmount of PolicyPV of tk. 1Net Single Premium3x4x51234561400.002831,0000.9712.747932410.003141,0000.9432.961023420.003501,0000.9153.202504430.003931,0000.8883.480965440.004391,0000.8633.78857Total =tk. 16.8098(because of rounding error)Net Single Premium in Whole Life PolicyA whole life policy continues for the whole life and promises to pay the sum assured upon the death of the insured to his beneficiary.It has been assumed in most of the mortality table that the life will continue up to 100 years. Therefore, the calculation of premium will start from the date of commencement of risk to 100 years.E.g., if a person takes a policy at age 45, the calculation will continue till 100th year. 16Example The chances of death in each separate year will be multiplied by the face value of the policy and this amount is discounted by the present value of the period.

17Table 10.4 Calculation of Net Single Premium on the basis of CSO Mortality TableAgeProbability of DeathPolicy amountPV of tk. 1 @2.5%=PV of the Claim457,340/852,5541,0000.975610=8.399441467,801/852,5541,0000.951814=9.039243..99125/852,5541,0000.257151=10.037703Net Single Premium (NSP)tk. 551.372587Net Single Premium in Pure Endowment PolicyIn this policy, insurer promises to pay the insured value in case the holder survives a certain fixed period. Thus, the holder of 5 year pure endowment will be paid only when he/she survives at the end 5 year. The insured cannot get possession of the money invested before the expiration of he endowment period. If the insured dies during this period, all the premium paid will be forfeited. 18Example 19Table 10.5 Calculation of Net Single Premium on the basis of Pure Endowment Actual form of CalculationProbability of SurvivalPolicy amountPV of tk. 1 for the Endowment Period=Net Single Premium=94,689/93,463=0.9811,0000.863=846.603Thus the NSP of tk. 1,000 for 5 years will be tk. 846.603

Net Single Premium in Ordinary Endowment PolicyUnder this policy, the payment of claim is made at the survival of the term or at the death of the life assured whichever is earlier.Payment in this case is certain.Since payment is based on the both death and survival, the net single premium is calculated on death and survival rate.

20Example Since the net single premium for 5-year term insurance and 5-year pure endowment insurance is already calculated, we can easily calculate the NSP for ordinary endowment policy as;NSP on the basis of death for 5 years = tk. 16.18NSP on the basis of survival at the end of 5 year = tk. 846.60tk. 862.78The equation used to calculate NSP for ordinary endowment is;21

Net Single Premium in Double EndowmentUnder this policy, double of the amount is paid if the life assured survives at the end of the term of policy and only single amount is paid if the death occurs within the term,Thus, it is first like ordinary endowment policy with only difference that double of the policy amount is paid if the life assured survives up to the term.Since the double of the policy amount is paid at the survival, one more premium on the basis of pure endowment is added to the premium of ordinary endowment policy. 22Example Double endowment policy is to be calculated of tk. 1,000 for 5 years. The net single premium of ordinary endowment + net single premium of pure endowment policy for 5 years at same age;=tk. 862.78+846.60=1,709.38The equation used to calculate double endowment policy premium is;23

Net Single Premium for a Joint Life PolicyUnder this policy, payment of claim will be made at the first death of the assured lives who may be two or more.Here, the process of calculation will be the same as has been shown in term insurance with only difference that the probability of death is compound one. The compound probability of death is calculated by addition of the probability of deaths of one other and all of the every aged. 24Example Compound probability of any one of the two lives assured will be;(a) Probability of death of the younger person +(b) Probability of death of the older person +(c) Probability of death of both the personsIt is calculated by multiplying the probability of death of each person. The following method is used to calculate the compound probability;

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Net Single Premium in AnnuitiesAn annuity is defined as a contract whereby, for a cash consideration (called premium), one party (the insurer) agrees to pay the other party (the annuitant) a stipulated sum (the annuity), throughout life, or during the life within a fixed term, either annually semiannually, quarterly, or monthly. The purpose of annuity is to protect a hazard the outliving of ones income.It is just opposite of insurance contract.The annuity protects against the absence of income in old age. A periodical amount is given by the insurance company to the insured up to his life or up to a fixed period of life thereafter.

26Net Single Premium in AnnuitiesUnder the annuity, the payment of annuity starts from the date of first annuity period.E.g., if annuity is to be paid six monthly, and the contract of annuity, i.e. payment of purchase price is made on 1st January, 2014, the first annuity will be paid on 1st June, 2014. Therefore, up to this period, the insurer can enjoy the premium paid six months ago.If the annuitant survives to next installment period, the insurer have to pay the second installment of the annuity.Annuity can be (1) Life Annuity, (2) Term Annuity.27Example Suppose net single premium for an immediate annuity of tk. 1,000 to be paid annually issued at the age of 105 is to be calculated.Assuming the rate of interest is 3 percent and the 1937 Standard Annuity Mortality Table is used.The calculation will be as follows;

28Table 10.6 Calculation of Net Single PremiumAgeProbability of SurvivalAmount of AnnuityPV of tk. 1 @ 3%=PV of Annuity10697/2491,0000.971=94,187/24910730/2491,0000.943=28,290/2491086/2491,0000.915=5,490/2491091/2491,0000.888=888/2491100/2491,0000.863=0/249Net Single Premium (NSP)tk. 128,855/249 = tk. 517Net Single Premium for Temporary AnnuityAn annuity of tk. 1,000 to be paid annually is taken at the age of 70 and continues up to 5 years. The rate of interest is 2.5% per annum.So the net single premium will b; 29Example 30Table 10.7 Calculation of Net Single PremiumAgeProbability of SurvivalAmount of AnnuityPV of tk. 1 @ 2.5%=PV of Annuity71553,332/577,4451,0000.975610=934.8772538,443/577,4451,0000.951814=871.0473502,843/577,4451,0000.928599=808.6374476,611/577,4451,0000.905951=747.7575449,841/577,4451,0000.883854=688.54tk. 4,050.79Equations

31Calculation of Level PremiumsThe present value (PV) per policy of all the net level premiums = net single premium of all the policyThe ratio between level premium and net single premium is 1:452,225/96,46332Table 10.7 Calculation of Net Single PremiumYears of InsuranceAgeNumber of Persons LivingNo. of Premium ReceivedAmount of Premium ReceivedPV of tk. 1PV of Premium Received14096,46396,46396,4631.00096,46324196,19096,19096,1900.97196,19034295,88095,88095,8800.94395,88044395,55295,55295,5520.91595,55254495,17795,17795,1770.88895,177Total present value=tk. 452,225Calculation of Level PremiumsOn this ratio the net single premium (NSP)of tk. 16.18 = 15,551,043/96,463 can be converted into several premium policy.

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Exercise # 7From the following mortality table, calculate the NSP of Term insurance of 5 years, Endowment insurance for 5 years, Pure endowment for 5 years, and Double endowment for 5 years. The rate of interest is 5 percentThe insurance amount is tk. 1,000.34x252627282930lx100,00099,50098,90098,20097,40096,500dx5006007008009001,000