Life Pricing Fundamentals Richard MacMinn

Objectives 2 January 2019 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. Copyright macminn.org

LAW OF LARGE NUMBERS 2 January 2019 The Weak Law of Large Numbers: For each n = 1, 2, . . ., suppose that R1, R2, . . . , Rn 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, Copyright macminn.org Robert Ash, Basic Probability Theory, Wiley 1970. See Theorem 2 on page 128. Take coins. Have each student flip the coin ten times and count the number of heads. Collect the results. Let a head be one and a tail be zero. Set R equal to one or zero as the coin toss yields a head or tail, respectively, on toss i = 1, 2, . . . . What do the sums represent? What is the expected value of S2? Note that S2 can be 0, 1 or 2 with probability ¼, 2/4, and ¼. Hence, the expected value is E(S2) = ¼ (0) + ½ (1) + ¼ (2) = 1 and E(S2)/2 = ½. Similarly, S3 can be 0, 1, 2, 3 with probability 1/8, 3/8, 3/8, and 1/8. Hence, the expected value is E(S3) = 1/8 (0) + 3/8 (1) + 3/8 (2) + 1/8 (3) = 12/8 = 1 ½ and E(S3)/3 = (1 ½)/3 = ½. Note that it follows by induction or direct calculation that E(Sn)/n = ½. What is your realized S2? S10? What was your S2/2? What was your S10/10? My S2 was 1 and so my S2/2 was ½. My S10 was 9 and my S10/10 was 9/10. What does the Weak Law of Large Numbers say about the probabilities of these results? What is the mean and standard deviation of this sample population?

Pricing objectives Adequacy Equity Not excessive 2 January 2019 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 Copyright macminn.org Excessive In some states a premium might be declared excessive if the insurer does not pay or expect to pay some specified percentage, e.g., 50%, of its premiums in claims.

Elements of rate making 2 January 2019 Probability of insured event Mortality and morbidity tables 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 Copyright macminn.org

Life 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 2 January 2019 Copyright macminn.org

SINGLE PREMIUM PLAN Table 2-2 Modified Version of 1980 CSO Mortality Table 1 2 3 4 Age Number Living (Beginning of Year) Probability of Death (During the Year) Number Dying (During the year) 95 100,000 0.330 33000 96 67,000 0.385 25795 97 41,205 0.480 19778 98 21,427 0.658 14099 99 7,328 1.000 7328 100 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 2 January 2019 Copyright macminn.org

Modified Version of 1980 CSO Mortality Table 2 January 2019 Copyright macminn.org

Present Value of Claims for 95-Year-Old Males 2 January 2019 Copyright macminn.org

Policy Reserves for Net Single-Premium Whole Life Insurance 2 January 2019 Copyright macminn.org

Level 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 pt be the proportion of the insured population alive at the beginning of policy year t. Let at 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. 2 January 2019 Copyright macminn.org

Net Level Premium Calculation 2 January 2019 Copyright macminn.org

Experience participation in insurance 2 January 2019 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. Copyright macminn.org

Asset share calculation 2 January 2019 The asset share calculation is a simulation of the anticipated operating experience of a block of policies An example Copyright macminn.org