LIFE PRICING FUNDAMENTALSRichard MacMinn

OBJECTIVES

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.

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LAW OF LARGE NUMBERS

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,

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PRICING 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

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ELEMENTS OF RATE MAKING

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

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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

Table 2-5Illustrative Net Level Premium Calculation

1 2 3 4 5

Policy Year

Net Level Premium to be Paid Annually

by Each Survivor

Number Living at the Beginning of

Each Year

Present Value Factor at 5%

Present Value of Total Net

Level Premiums [(2)

x (3) x (4)]

1 1 100,000 1.0000 $100,0002 1 67,000 0.9524 $63,8103 1 41,205 0.9070 $37,3744 1 21,427 0.8638 $18,5095 1 7,328 0.8227 $6,029

Total PV $225,722

premium $395.40

Richard:This is the level premium or premium per year.

The $225,722 is the present value per dollar in premiums paid each year of the policy. Hence, that times the premium per year must equal the present value of the claims, i.e., the $89,251,339. By altering the interest rate in table 2-3 cell C1 is possible to see how the level premium changes in table 2-5 cell E12

Richard D. MacMinn:This allows us to calculate the present value of a one dollar premium flow per customer.

April 10, 2023 Copyright macminn.org6

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

1 2 3 4

AgeNumber 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 0

April 10, 2023 Copyright macminn.org7

MODIFIED VERSION OF 1980 CSO MORTALITY TABLE Table 2-5Illustrative Net Level Premium Calculation

1 2 3 4 5

Policy Year

Net Level Premium to be Paid Annually

by Each Survivor

Number Living at the Beginning of

Each Year

Present Value Factor at 5%

Present Value of Total Net

Level Premiums [(2)

x (3) x (4)]

1 1 100,000 1.0000 $100,0002 1 67,000 0.9524 $63,8103 1 41,205 0.9070 $37,3744 1 21,427 0.8638 $18,5095 1 7,328 0.8227 $6,029

Total PV $225,722

premium $395.40

Richard:This is the level premium or premium per year.

The $225,722 is the present value per dollar in premiums paid each year of the policy. Hence, that times the premium per year must equal the present value of the claims, i.e., the $89,251,339. By altering the interest rate in table 2-3 cell C1 is possible to see how the level premium changes in table 2-5 cell E12

Richard D. MacMinn:This allows us to calculate the present value of a one dollar premium flow per customer.

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PRESENT VALUE OF CLAIMS FOR 95-YEAR-OLD MALESTable 2-5Illustrative Net Level Premium Calculation

1 2 3 4 5

Policy Year

Net Level Premium to be Paid Annually

by Each Survivor

Number Living at the Beginning of

Each Year

Present Value Factor at 5%

Present Value of Total Net

Level Premiums [(2)

x (3) x (4)]

1 1 100,000 1.0000 $100,0002 1 67,000 0.9524 $63,8103 1 41,205 0.9070 $37,3744 1 21,427 0.8638 $18,5095 1 7,328 0.8227 $6,029

Total PV $225,722

premium $395.40

Richard:This is the level premium or premium per year.

The $225,722 is the present value per dollar in premiums paid each year of the policy. Hence, that times the premium per year must equal the present value of the claims, i.e., the $89,251,339. By altering the interest rate in table 2-3 cell C1 is possible to see how the level premium changes in table 2-5 cell E12

Richard D. MacMinn:This allows us to calculate the present value of a one dollar premium flow per customer.

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POLICY RESERVES FOR NET SINGLE-PREMIUM WHOLE LIFE INSURANCETable 2-5Illustrative Net Level Premium Calculation

1 2 3 4 5

Policy Year

Net Level Premium to be Paid Annually

by Each Survivor

Number Living at the Beginning of

Each Year

Present Value Factor at 5%

Present Value of Total Net

Level Premiums [(2)

x (3) x (4)]

1 1 100,000 1.0000 $100,0002 1 67,000 0.9524 $63,8103 1 41,205 0.9070 $37,3744 1 21,427 0.8638 $18,5095 1 7,328 0.8227 $6,029

Total PV $225,722

premium $395.40

Richard:This is the level premium or premium per year.

The $225,722 is the present value per dollar in premiums paid each year of the policy. Hence, that times the premium per year must equal the present value of the claims, i.e., the $89,251,339. By altering the interest rate in table 2-3 cell C1 is possible to see how the level premium changes in table 2-5 cell E12

Richard D. MacMinn:This allows us to calculate the present value of a one dollar premium flow per customer.

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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.

Tt

T t 1t 1

1 2 3 2

4 53 4

pa

(1 r)

1 1p p p

1 r 1 r

1 1p p

1 r 1 r

T Ta x pv (L)

April 10, 2023 Copyright macminn.org11

NET LEVEL PREMIUM CALCULATIONTable 2-5Illustrative Net Level Premium Calculation

1 2 3 4 5

Policy Year

Net Level Premium to be Paid Annually

by Each Survivor

Number Living at the Beginning of

Each Year

Present Value Factor at 5%

Present Value of Total Net

Level Premiums [(2)

x (3) x (4)]

1 1 100,000 1.0000 $100,0002 1 67,000 0.9524 $63,8103 1 41,205 0.9070 $37,3744 1 21,427 0.8638 $18,5095 1 7,328 0.8227 $6,029

Total PV $225,722

premium $395.40

Richard:This is the level premium or premium per year.

The $225,722 is the present value per dollar in premiums paid each year of the policy. Hence, that times the premium per year must equal the present value of the claims, i.e., the $89,251,339. By altering the interest rate in table 2-3 cell C1 is possible to see how the level premium changes in table 2-5 cell E12

Richard D. MacMinn:This allows us to calculate the present value of a one dollar premium flow per customer.

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EXPERIENCE 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.

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ASSET SHARE CALCULATION

The asset share calculation is a simulation of the anticipated operating experience of a block of policies

An example

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