# L IFE P RICING F UNDAMENTALS Richard MacMinn. O BJECTIVES Understand the law of large numbers as it relates to insurance. Describe insurers pricing objectives.

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L IFE P RICING F UNDAMENTALS Richard MacMinn

O BJECTIVES Understand the law of large numbers as it relates to insurance. Describe insurers pricing objectives and explain why they are of relevance to the life insurer and consumer. Outline elements of life insurance rate making including the assumptions made in the absence of perfect information. Draw distinctions between participating and guaranteed cost, nonparticipating life insurance. Explain how asset share analysis is used to test the adequacy of life insurance rates. 1 June 2014 2 Copyright macminn.org

LAW OF LARGE NUMBERS The Weak Law of Large Numbers: For each n = 1, 2,..., suppose that R 1, R 2,..., R n are independent random variables on a given probability space, each having finite mean and variance. Assume that the variances are uniformly bounded; that is, assume that there is some finite positive number M such that for all i. Let Then, 1 June 2014 3 Copyright macminn.org

P RICING OBJECTIVES Adequacy The payments generated by a block of policies plus any investment return on same must be sufficient to cover the current and future benefits and costs Equity This equity refers to setting premiums commensurate with the expected losses and expenses; it also suggests no cross subsidization. The equity notion sets a floor. Not excessive The excessive notion sets a ceiling Regulation Competition 1 June 2014 4 Copyright macminn.org

E LEMENTS OF RATE MAKING Probability of insured event Mortality and morbidity tables Mortality Time value of money Premiums paid now Interest on accumulated funds Promised benefit period of coverage level of coverage type of coverage Loading or expenses, taxes, contingencies and profit 1 June 2014 5 Copyright macminn.org

L IFE INSURANCE RATE COMPUTATION Yearly renewable term life insurance The YRT covers the life for one year at a set premium and is renewable The YRT premium for a 30 year old male would be \$1.73 per \$1,000 of coverage while it would be \$1.38 for a female the same age. If investment income is included then the company would set the premium at \$1.65 and \$1.31 for males and females respectively Single premium plan Level premium plan 1 June 2014Copyright macminn.org 6

SINGLE PREMIUM PLAN This plan provides multi- year coverage for a single premium now This eliminates the rising premiums associated with the YRT. This gives the insurer the ability to generate compound interest and reduce the rate for coverage Table 2-2 Modified Version of 1980 CSO Mortality Table 1234 Age Number Living (Beginning of Year)of Probability of Death (During the Year) Number Dying (During the year) 95100,0000.33033000 9667,0000.38525795 9741,2050.48019778 9821,4270.65814099 997,3281.0007328 1000 1 June 2014Copyright macminn.org 7

M ODIFIED V ERSION OF 1980 CSO M ORTALITY T ABLE 1 June 2014 8 Copyright macminn.org

P RESENT V ALUE OF C LAIMS FOR 95-Y EAR - O LD M ALES 1 June 2014 9 Copyright macminn.org

P OLICY R ESERVES FOR N ET S INGLE - P REMIUM W HOLE L IFE I NSURANCE 1 June 2014 10 Copyright macminn.org

L EVEL PREMIUM PLAN If some of the 100,000 policyholders prefer to pay premiums on an annual basis then how much must be charged per year to make the insurer indifferent between the single premium and the annual level premium? Let p t be the proportion of the insured population alive at the beginning of policy year t. Let a t be the annuity factor for the premium payment stream. Let x be the level premium. Then x must satisfy the last equation on the RHS. 1 June 2014Copyright macminn.org 11

N ET L EVEL P REMIUM C ALCULATION 1 June 2014 12 Copyright macminn.org

E XPERIENCE PARTICIPATION IN INSURANCE Guaranteed-cost, non-participating insurance (without profits policies) Policy elements fixed at inception They offer no way of passing changes in mortality (morbidity), interest or loading to policyholders Participating insurance (with profits policies) Policy gives its owner the right to share in surplus accumulated due to experience Surplus is distributed as dividends Current assumption insurance Policy allows values to deviate from those at policy inception on the upside and downside Unlike participating policies that adjust ex post the current assumption policy adjusts ex ante; for example, if the insurer expects a 7% return on investments backing policy reserves then the policyholders may get a promised 6.5% credited to their cash values. 1 June 2014 13 Copyright macminn.org

A SSET SHARE CALCULATION The asset share calculation is a simulation of the anticipated operating experience of a block of policies An exampleexample 1 June 2014 14 Copyright macminn.org

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