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Actuarial Science from the Air An Overview of Life Contingencies, Including Life Annuities and Life Insurance Introduction From the ground, individual streams may seem isolated; but from the air they are visible as interconnected features of a larger watershed. Similarly, in the study of Actuarial Science, a number of apparently dissimilar insurance products actually have a strong theoretical relationship. The unifying principles that provide this broad view are now explained and applied. Properties of Insurance Products • A policy can be in effect for a certain term or for the whole life of the insured • Benefits can begin immediately or be deferred • Life insurance pays upon death • Life annuities provide a series of payments contingent upon survival While Life insurance and Life annuities differ, the same three unifying principles apply when determining their costs. Unifying Principle #1: Invested money grows over time. Simplifying Assumption #1: All investments grow at an annual compound interest rate of 5%. Example 1 (Present Value): To accumulate $1000 in 10 years, invest $613.91 because $613.91(1.05)10 = $1000 The amount $613.91 is called the present value of $1000 to be paid in 10 years. Example 2: To receive $1000 at the end of each year for the next 10 years, invest $7,721.73, the sum of the 10 present values: Present Values for a Sequence of Ten $1000 Payments Year Present Values 1 $952.38 2 $907.03 3 $863.84 4 $822.70 5 $783.53 6 $746.22 7 $710.68 8 $676.84 9 $644.61 10 $613.91 SUM = $7,721.73 Accumulated Values $952.38 (1.05)1 = $1,000 2 $907.03 (1.05) = $1,000 3 $863.84 (1.05) = $1,000 4 $822.70 (1.05) = $1,000 5 $783.53 (1.05) = $1,000 6 $746.22 (1.05) = $1,000 7 $710.68 (1.05) = $1,000 $676.84 (1.05)8 = $1,000 9 $644.61 (1.05) = $1,000 10 $613.91 (1.05) = $1,000 SUM = $10,000 Present Value Formula The present value P of an amount X to be paid after n years (at 5% interest) satisfies the following equivalent equations P(1.05)n = X P = X/(1.05)n Example 3: Other Present Values Year PresentValues PresentValues PresentValues of X of $1 of $1000 0.95238X $0.95238 $952.38 1 0.90703X $0.90703 $907.03 2 0.86384X $0.86384 $863.84 3 0.82270X $0.82270 $822.70 4 0.78353X $0.78353 $783.53 5 0.74622X $0.74622 $746.22 6 0.71068X $0.71068 $710.68 7 0.67684X $0.67684 $676.84 8 0.64461X $0.64461 $644.61 9 0.61391X $0.61391 $613.91 10 Unifying Principle #2: The cost of providing insurance benefits depends upon when the insured dies. For example, the longer a person lives, • the smaller the present value of his life insurance payment (decreasing the cost) • the greater the number of his life annuity payments (increasing the cost) Life Table In determining what to charge for insurance, a company needs to know the likelihood or probability of death at various times. This information may be given in a life table. Simplifying Assumption #2 Uniform Distribution of Deaths: Of the people alive at age 75, 5% of them will die in each of the next 10 years. For example, if 10,000 are alive at age 75, then • 500 (= 5% of 10,000) will die each year • 2000 (= 4(500)) will die in the next 4 years • 8000 (= 10,000 – 2000) will survive the next 4 years • 0.80 = 8000/10000 = probability of surviving the next 4 years Year n 1 2 3 4 5 6 7 8 9 10 Simplified Life Table In Year n End of Year n Probability Number of Number Probability of of death deaths alive survival 0.05 500 9500 0.95 0.05 500 9000 0.90 0.05 500 8500 0.85 0.05 500 8000 0.80 0.05 500 7500 0.75 0.05 500 7000 0.70 0.05 500 6500 0.65 0.05 500 6000 0.60 0.05 500 5500 0.55 0.05 500 5000 0.50 Cost of a Future Benefit Problem 1: Suppose each of two 75-year-old men wishes to receive a benefit of $1000 if he is still living after 10 years. How much should each pay now to receive that future benefit? • • • • • The probability of surviving 10 years is 0.50. On average, only one of the two will survive 10 years. On average, only one benefit of $1000 will be paid. The present value of this benefit is $613.91. Thus, the men split this cost: Cost for each = $613.91 / 2 = $306.96 Second Solution to Problem 1: A 75-year-old man has a 50% chance of surviving 10 years. Therefore, the cost of paying him $1000 if he is alive after 10 years is 50% of the present value of the payment: Cost = [$1000/(1.05)10] 0.50 = [$613.91] 0.50 = $306.96 Unifying Principle #3 Suppose a payment A will be made with probability p after n years. Then • The cost varies directly as p • More precisely, the cost equals the present value of the payment times p: Cost = [A/(1.05)n] p (This cost is known as the actuarial present value of the payment A.) Computing Costs by Unifying Principle #3 We now compute the costs of • a 10-year term life annuity that pays $1000 at the end of each year of survival for the next 10 years • 10-year term life insurance that pays $1000 at the end of the year of death if death occurs within 10 years These are alike in that both offer a possible payment at the end of each of the next ten years. Thus, by Unifying Principle #3 the cost of each is the sum of 10 actuarial present values. Costs of 10-year Term Life Annuity and 10-year Term Life Insurance Each with Potential Payments of $1000 Year n Present value Life Annuity Life Insurance of $1000 to be Probability Cost of Probability Cost of paid in n years of payment payment of payment payment 1 $952.38 0.95 $904.76 0.05 $47.62 2 $907.03 0.90 $816.33 0.05 $45.35 3 $863.84 0.85 $734.26 0.05 $43.19 4 $822.70 0.80 $658.16 0.05 $41.14 5 $783.53 0.75 $587.64 0.05 $39.18 6 $746.22 0.70 $522.35 0.05 $37.31 7 $710.68 0.65 $461.94 0.05 $35.53 8 $676.84 0.60 $406.10 0.05 $33.84 9 $644.61 0.55 $354.53 0.05 $32.23 10 $613.91 0.50 $306.96 0.05 $30.70 TOTALS: $7,722 COST = $5,753 COST = $386 5-year Term Life Annuities (undeferred) 5-year term pays at the end of each year of survival for years 1-5. The probability of payment equals 0 for years 6-10. 5-year deferred, 5-year term pays at the end of each year of survival for years 6-10. The probability of payment equals 0 for years 1-5. The following table applies Unifying Principle #3 to compute the two costs: n 1 2 3 4 5 6 7 8 9 10 Cost of 5-year Term Life Annuities Present value Undeferred 5-year Deferred of $1000 to be Probability Cost of Probability Cost of paid in n years of payment payment of payment payment $952.38 0.95 $904.76 0 $907.03 0.90 $816.33 0 $863.84 0.85 $734.26 0 $822.70 0.80 $658.16 0 $783.53 0.75 $587.64 0 $746.22 0 0.70 $522.35 $710.68 0 0.65 $461.94 $676.84 0 0.60 $406.10 $644.61 0 0.55 $354.53 $613.91 0 0.50 $306.96 COST = $3701 COST = $2052 TOTALS: Final Problem Problem 2: What yearly premium X, paid at the beginning of each year for 10 years, would cover the cost of 10 year term life insurance for a person aged 75? Simplifying Assumption #3 Equivalence Principle: The cost to the insured equals the cost to the insurance company. That is, the following are equal: • Actuarial present value of the 10 premium payments of X • Actuarial present value of the benefit payment We now compute these expected present values and use their equality to solve for X: Yearly Premium X for 10-year Term Life Insurance Paying $1000 Life Insurance Premium Payments n Pres value Prob of Cost of Pres value Prob of Cost of of $1000 pd payment payment of X paid payment premium in n years in n years 0 X 1.00 X 1 $952.38 0.05 $47.62 0.95238X 0.95 0.90476X 2 $907.03 0.05 $45.35 0.90703X 0.90 0.81633X 3 $863.84 0.05 $43.19 0.86384X 0.85 0.73426X 4 $822.70 0.05 $41.14 0.82270X 0.80 0.65816X 5 $783.53 0.05 $39.18 0.78353X 0.75 0.58764X 6 $746.22 0.05 $37.31 0.74622X 0.70 0.52235X 7 $710.68 0.05 $35.53 0.71068X 0.65 0.46194X 8 $676.84 0.05 $33.84 0.67684X 0.60 0.40610X 9 $644.61 0.05 $32.23 0.64461X 0.55 0.35453X 10 $613.91 0.05 $30.70 Equal Totals Add the columns: $386.09 6.44607X Yearly Premium = X = $386.09/6.44607 = $59.90 Homework Problem 1 How would each of the following affect the cost of life annuities and life insurance? • Change the annual interest rate from 5% to 3% • Change the number of deaths in the life table from 500 to 600 per year Homework Problem 2: Complete the following more realistic life table: Year n 1 2 3 4 5 6 7 8 9 10 Life Table for 10,000 Alive at Age 75 In Year n End of Year n Probability Number of Number Probability of of death deaths alive survival 451 465 479 494 509 523 534 542 543 537 Homework Problem 3 Assuming 6% annual compound interest, use the life table from Problem 2 to compute the costs of a 10-year term life annuity and 10-year term life insurance. Conclusion In a brief presentation, we have • introduced important actuarial science concepts • seen a unified discussion of life insurance and life annuities