# THE MATHEMATICS OF LIFE INSURANCE THE NET SINGLE PREMIUM. - Investigate the component parts of the life insurance premium.

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1 THE MATHEMATICS OF LIFE INSURANCE THE NET SINGLE PREMIUM CHAPTER OBJECTIVES - Discuss the importance of life insurance mathematics. - Investigate the component parts of the life insurance premium. - Demonstrate how the net single premium is calculated for traditional life insurance contracts. - Demonstrate how the net single premium is calculated for annuity contracts. INTRODUCTION No study of life insurance would be complete without an examination of the mathematics used in calculating the insurance premium. Mathematics allows the accurate calculation of life insurance premiums and provides the foundation for the life insurance transaction. The amount of the premium, the contractual obligations, the investment return, the mortality experience, expenses, and taxes all shape and mold the balance sheet and income statements of life insurance companies. The background material and the method for calculating the net single premium will be presented in this chapter. The net single premium is the one-time charge the insurance company requires the policyholder to pay in order to fulfill the promise specified in the contract. The premium is referred to as net because it assumes payment is only for mortality costs. No expenses of providing the insurance contract (such as taxes, overhead, and additions to surplus) are included. Single denotes that the premium will be paid only once. The policyowner need not pay another premium, and the insurer will remain obligated to pay all benefits

2 Knowing how to calculate the net single premium is the first step in understanding the mathematics of life insurance. In subsequent chapters additional mathematical topics will be developed. The topics include: - calculating a level amount of premium to be paid each period (the level premium); - policy reserve values (a balance sheet liability); and - the gross premium (the premium that includes taxes, overhead, and additions to surplus). THE MATHEMATICS OF LIFE INSURANCE There are three reasons for studying the mathematics of life insurance. Perhaps the most important is that life insurance contracts call for monetary payments from the insurer, usually a long time after the policy begins. Thus the insurance company needs to calculate very accurately the amount of money needed to deliver its future promises to policyholders. Underpricing the contract will ultimately lead to losses. Overpricing the contract can lead to reduced sales due to competitive pressures. Fewer policies sold at a higher price is undesirable, other factors being the same, because the life insurance company's mortality predictions will be less accurate and will in turn increase the variability of operating results. 1 Thus serious mistakes in pricing the life insurance product can harm the life insurance company's profit picture and threaten its existence. The second reason for studying life insurance mathematics is that the premium charged is a function of the event insured. Mathematical models are used to predict the relative frequency of the insured event and to calculate the premium required to insure that event. It is important to know how these models (such as a mortality table) are 1 The law of large numbers states that the larger the sample size the closer will the actual approach the expected experience of the group.

3 developed in order to better understand the calculation of the life insurance premium. If the insured event is death, the price (premium) is a function of mortality, or death, rates. Health and disability premiums are dependent upon morbidity rates, and life annuity contracts are dependent upon survivor rates, which are the inverse of mortality rates. 2 The third reason for studying life insurance mathematics is that the contracts often last for many years (obligations of more than 100 years are possible when settlement options are considered) and the time value of money is very important. It is well established that the value of one dollar received today is worth more than one dollar received 10 years from now. The time value of money comes into play in life insurance mathematics in many ways. Money received can be invested for long periods of time until funds are needed to pay claims. During this accumulation phase, the insurance company relies on compound interest to provide sufficient funds to pay the benefits called for by the contract. If a lump-sum settlement is not chosen at death, the investment rate also influences the amount of proceeds paid to the beneficiary. For example, the amount or length of payments under the fixed-period, fixed- amount, and life annuity options partially depends on the rate of return earned on the invested funds. It is clear from this brief discussion that the basis for operating a successful life insurance company is the calculation of the proper insurance rate; that is, the company's success is dependent upon accurate mathematics. Therefore it is evident that once a life insurance contract is sold, the initial premium charged will continue to have a dynamic impact on the company's financial statements and well being throughout the life of the contract. COMPOSITION OF THE PREMIUM 2 Different tables are used however because the experience of groups of people seeking life insurance is different than groups seeking annuity contracts.

4 The life insurance premium is a function of four factors: - the time value of money, - the probability of the insured event occurring, - the amount to be paid if the insured event occurs, and - the amounts added to the premium for expenses, additions to surplus, and taxes. The gross premium needed to complete the contractual promise(s) can be calculated only when all of these factors are considered. The first part of this calculation focuses on the net single premium which is derived from considering the amount paid when the event occurs, the time value of money, and the probability of the event. Each of these factors will be explored in turn. Amounts added to the premium for expenses, additions to surplus and taxes are considered in subsequent chapters. The Time Value of Money: The first Component The impact of compound interest on the life insurance premium calculation is significant when long contractual periods are involved. The time value of money is one of the important building blocks of the premium. If time value of money concepts are unfamiliar or difficulty is encountered in making the calculations, the material in Chapter 13 should be reviewed. Mortality: The Second Component The probability of loss is the second component considered when the premium is calculated. The probability of loss is the expected frequency of the loss occurring and is stated as a fraction. For example, if 100 individuals are expected to die out of a group of 10,000, the expected probability is 0.01 (100 / 10,000). In life insurance, mortality rates are dependent upon many factors. The most important factor is age. A group of 20-year-olds will live longer than a group of 60-year-

5 olds, though some 60-year-olds will outlive some 20-year-olds. Other significant factors are sex, health, lifestyle, hazardous occupations or hobbies, and the length of time the person has been insured. In calculating the probability of loss, companies must consider, on the average, that females tend to live longer than males; healthy people tend to live longer than the unhealthy; and people who do not drink, smoke, and engage in hazardous occupations or hobbies also tend to live longer. Newly insured groups experience a lower mortality rate than groups insured for several years and are the same age. Males age 35 who have just successfully passed a life insurance medical exam will generally show a lower mortality rate than men age 35 who have been insured for several years. The lower mortality rate for the new group is a result of the use of a medical examination for preselection purposes. This is known as the medical selection effect. Life insurance companies use mathematical models for predicting death rates. There are many mathematical models describing a function that can predict the number of survivors of a cohort (group) of lives over time. One job of the life insurance actuary is to select a mathematical model (mortality table) that reflects accurately the mortality rates the life insurance company expects to experience. Thus a life insurance company must select or create a mortality table reflecting expected experience so that accurate predictions may be obtained. Continuous mathematical models, even though they produce smooth curves, do not reflect the mortality rates accurately at all ages; therefore the parameters of the mathematical model, coefficients that shape the function, must be refined so mortality or morbidity predictions are generally accurate when large numbers of insureds are considered in the pool. Life insurance companies also can construct mortality or morbidity tables by selecting a large group of individuals and observing its mortality rate. A large cohort of lives is selected for observation at a certain age (usually age zero for life insurance), and the experience of the group is tabulated over many years until there are no

6 survivors. If 10 million male lives were selected for observation at age zero, accurate statistics could be developed by following the cohort and dividing the number of people who die during each year by the number of people who began that year, to derive the frequency or probability of death. If the radix (the number of individuals that start the group) is large, one would assume that the rate of deaths per year would change in a smooth fashion; but irregularities do occur. A process of graduation is used to smooth any apparent inconsistencies in the annual experience, so that the tabular rates are closer to the theoretical rates. To illustrate how to construct and interpret a mortality table, the Commissioners 1980 Standard Ordinary (CSO) Table (Table 1) will be used in this chapter. 3 The 1980 CSO Table will also be used to illustrate the calculation of life insurance premiums in the material to follow. Interpreting mortality tables. The 1980 CSO Table starts at age zero (x) with 10 million male lives as its radix. 4 Out of the 10 million lives that start the year (l x ), it is expected that 41,800 will die (d x ) during the year, leaving 9,958,200 to start the next year. The probability of a person dying during the year (q x=0 ) is equal to d 0 / l 0 = 41,800 / 10,000,000 or deaths per thousand. Similarly the yearly probability of dying during age 40 (q 40 ) is equal to d 40 / l 40 = 28,319 / 9,377,225 or deaths per thousand. 3 The 1980 CSO mortality table and the 1983 Annuity Tables for male and female lives are found in the appendix to this chapter. The 1980 CSO Table is based on experience in 1970 to The 1983 Annuity Table is based on 1971 to 1976 experience. 4 In 1898 the International Congress of Actuaries established a system of notation for actuarial literature. The following notation will be used where convenient in this and the following chapters: x = age; q x = probability of dying during age x (d x / l x ); px = probability of surviving age x (l x+1 / l x ); lx = number living at age x (l x-1 - d x-1 ); dx = the number dying during age x (l x - l x+1 ).

7 TABLE 1 Commissioners 1980 Standard Ordinary ( ) Male lives Number Number Yearly Yearly alive at the start dying during the probability of probability of Age of the year year dying surviving l x d x q x p x 0 10,000,000 41, ,958,200 10, ,947,545 9, ,937,697 9, ,927,958 9, ,918,526 8, ,909,599 8, ,901,077 7, ,893,156 7, ,885,637 7, ,878,322 7, ,871,111 7, ,863,510 8, ,855,126 9, ,845,369 11, ,834,047 13, ,806,138 16, ,789,762 17, ,772,336 18, ,754,159 18, ,735,626 18, ,717,031 18, ,698,666 18, ,680,626 17, ,663,007 17, ,645,903 16, ,629,216 16, ,612,750 16, ,596,408 16, ,579,998 16, ,563,425 17, ,546,402 17, ,528,932 18, ,510,732 19, ,491,711 20, ,471,683 21, ,450,466 22, ,427,785 24, ,403,461 26,

8 40 9,377,225 28, ,348,906 30, ,318,148 33, ,284,975 35, ,249,042 38, ,210,289 41, ,168,382 45, ,123,274 48, ,074,738 52, ,022,649 56, ,966,618 60, ,906,452 65, ,841,435 70, ,771,057 76, ,694,661 83, ,611,540 90, ,521,377 97, ,423, , ,318, , ,205, , ,084, , ,954, , ,814, , ,664, , ,503, , ,329, , ,143, , ,944, , ,733, , ,509, , ,274, , ,026, , ,765, , ,490, , ,201, , ,898, , ,584, , ,261, , ,932, , ,602, , ,274, , ,950, , ,633, , ,324, , ,026, ,

9 85 1,742, , ,475, , ,230, , ,009, , , , , , , , ,472 97, ,281 77, ,381 61, ,721 48, ,309 37, ,504 29, ,450 20, ,757 10, And the yearly probability of dying between age 99 and 100 (q 99 ) is d 99 / l 99 = 10,757 / 10,757 = 1.00 Thus the 1980 CSO mortality table assumes no one survives to age 100. The yearly probability of surviving (p x is simply unity (1) minus the yearly probability of dying. p x = 1 - q x Calculating the probability of dying or surviving for a period of years is a relatively easy task, using the mortality tables supplied. For example, for a male age 35, what is the probability of dying within the next 10 years and the probability of surviving the 10- year period? (Use the 1980 CSO table.) The probability of dying within the next 10-year period is equal to the number of people who die within the 10-year period (l 35 - l 45 ) divided by the number of people starting the cohort at age 35 (l 35 ). (l 35 - l 45 ) 9,491,711-9,210,289 q = = = l 35 9,491,711

10 Similarly the probability of surviving the 10-year period is equal to the number of people who survive the 10-year period divided by the number of people who started the cohort. l 45 9,210,289 p = --- = = l 35 9,491,711 or p = = The number living is taken from age 45 because l 45 is the number of people alive at the beginning of age 45, which is equal to the number of individuals alive at the end of age 44. One must be careful in interpreting the above probabilities. Predictions of mortality are only accurate when applied to a large number of lives. One might argue that it is erroneous to state that the probability of death of a particular male age 40 is since one male is an insufficient number to base any credible predictions upon. On the other hand, it is accurate to say that out of 100,000 men age 40, it is expected that 302 of them will die during the year. It is impossible to make accurate predictions based on a group consisting of one individual. The 1980 CSO Table is a conservative table in that the death rates of insured lives were overstated initially and is based on 1970 to 1975 data. Because death rates have declined since the table was developed, the 1980 CSO Table is even more overstated. Life insurance companies are not required to use any particular mortality table for calculating their rates. However, life insurance companies are required to use

11 the 1980 CSO Table for calculating the legal reserves on their balance sheet, thus imposing a form of indirect rate regulation. 5 Annuity tables. Since people who purchase annuities exhibit better mortality rates (fewer deaths per thousand) than people who purchase life insurance contracts, different tables must be constructed for use with annuity contracts than the ones used in life insurance contracts. The 1983 Annuity Table with Projection (Table 2) is commonly used for predicting mortality among a group of individuals purchasing annuities. The table is similar to the 1980 CSO Table except that the annuity table starts at age 5 and every annuitant is assumed to die before age 116. The 1983 Annuity Table is based on 1971 to 1976 data and is interpreted in a similar fashion as the 1980 CSO Mortality Table. For a comparison of several mortality tables refer to Table 3. Many annuity tables are labeled "projected." A projected table predicts a lower mortality rate (higher survivor rate) than similar tables that are not projected. Projected tables are derived by taking the mortality rates predicted by nonprojected tables and adjusting them for anticipated mortality trends. As mortality rates decline, annuity tables become less conservative. If mortality is less than predicted (longevity increases), more money than anticipated must be paid in rent, and the life insurance company could experience financial loss. Declining mortality rates in life insurance contracts are not a problem; if fewer people die than expected, the life insurance company will not be threatened financially. Types of mortality tables. As stated, mortality rates differ between newly insured individuals and those insured for several years. The effect of this selection process eventually disappears, and the mortality rates of the two groups at the same age will not 5 As of this writing, all states have adopted the 1980 CSO Table for calculating reserves. The death rates predicted in the 1980 CSO Table are generally lower than the death rates predicted by the previously used 1958 CSO Table.

12 differ significantly. Mortality tables showing this ripple effect from the age of issue to a later date (three to four years) are called select mortality tables. The mortality rate is dependent first upon the age of the group and second on the number of years since the group has gone through the insurance company's selection process. Thus a lower mortality rate is experienced for a select group age 35 versus a group age 35 insured for a short period. When the effect of the selection finally erodes, the mortality rate for a male age 35 who has been insured for five years is approximately equal to one insured for many years. As the number of years being insured increases, the select mortality rates converge into the mortality rates of groups insured for many years; thus the select mortality rates will evolve into ultimate mortality rates. The 1980 CSO table and the 1983 Annuity Table are examples of ultimate mortality tables. Age TABLE 2 Commissioners 1983 Individual Annuity Table ( Projected to 1980) Male lives Number alive at the start of the year Number dying during the year Yearly probability of dying Yearly probability of surviving l x d x q x p x 5 10,000,000 3, ,996,230 3, ,992,731 3, ,989,403 3, ,985,887 3, ,982,212 3, ,978,399 3, ,974,468 4, ,970,428 4, ,966,290 4, ,962,054 4, ,957,721 4, ,953,280 4, ,948,721 4, ,944,025 4, ,939,172 5, ,934,153 5, ,928,938 5,

13 23 9,923,517 5, ,917,861 5, ,911,950 6, ,905,785 6, ,899,346 6, ,892,644 6, ,885,680 7, ,878,454 7, ,870,956 7, ,863,197 8, ,855,168 8, ,846,860 8, ,838,234 9, ,829,212 9, ,819,697 10, ,809,563 10, ,798,635 11, ,786,720 13, ,773,596 14, ,759,014 16, ,742,687 18, ,724,312 20, ,703,609 23, ,680,330 26, ,654,261 29, ,625,211 32, ,593,034 35, ,557,597 38, ,518,822 42, ,476,644 45, ,431,042 49, ,382,019 52, ,329,564 55, ,273,643 59, ,214,208 63, ,151,192 66, ,084,480 70, ,013,785 75, ,938,628 80, ,858,332 86, ,772,052 93, ,678, , ,577, , ,467, , ,347, ,

14 68 8,215, , ,072, , ,917, , ,747, , ,564, , ,366, , ,154, , ,927, , ,684, , ,426, , ,152, , ,863, , ,560, , ,243, , ,913, , ,574, , ,227, , ,875, , ,522, , ,173, , ,832, , ,502, , ,188, , ,893, , ,619, , ,367, , ,139, , , , , , , , , , ,864 90, ,218 72, ,285 57, ,948 43, ,065 32, ,517 23, ,255 15, ,346 10, ,021 6, ,727 3, ,168 1, ,

15 TABLE 3 Selected Mortality Table Statistics Deaths per thousand U.S. Age Mortality Mortality Annuity Annuity Population All male lives except U.S. Population 1949 Annuity Table is not projected 1983 Individual Annuity Table (based on experience) projected to CSO Table (based on experience) and increased to produce conservative statistics.

16 FIGURE 1 CALCULATING THE NET SINGLE PREMIUM The net single premium (NSP) equals the amount of money the life insurance company must have on hand at the beginning of the contract term in order to perform according to its contractual obligations. The net single premium does not include any amounts for taxes, operating expenses, or additions to surplus and therefore is not representative of the premium a consumer will pay. But it is a very useful teaching device that has actuarial applications. Calculating the net single premium involves determining the present value of the benefits to be provided by the contract. This statement will become clearer as the calculation process is presented. NSP Assumptions

17 The following presentation of the mathematics underlying the net single premium assumes several unrealistic assumptions in order to simplify the calculation process. In order to have the net single premium equal the present value of the future benefits, these assumptions are made: - The group insured is large enough to produce absolutely predictable results. In other words, the mortality rate of the group insured is equal to the mortality rate predicted by the table being used (in this case the 1980 CSO Table). 6 - All premiums are collected at exactly the same instant at the beginning of the contract year. - All death claims are paid at the end of the policy year. In practice, actuarial calculations start with the above simplifying assumptions and use various techniques to handle problems generated by grouping individuals with unequal birth dates in a given year and collecting premiums and paying death claims throughout the policy year. The reader should evaluate the effect on the net single premium when there are violations of one or more of the above assumptions. The Term Insurance Contract Term life insurance contracts promise to pay \$1,000 to a beneficiary if the insured dies within the term specified. 7 Term insurance contracts may last for many 6 An implied extension to this assumption is that all insureds are the same age and sex, and all are in good health. If unisex mortality tables are used (mortality tables that combine both males and females in one group), males and females may be mixed in the same proportion used in constructing the table. The result regardless must be absolutely predictable. 7 It will be assumed that life insurance contracts are issued only for \$1,000 for illustrative purposes. Life insurance policies are purchased for any practical amount. The premium calculation is similar except that the benefit changes or the resulting net single premium may be multiplied by the face amount desired in thousands.

18 years but are considered temporary contracts since an individual generally cannot renew the contract past age 65 to The net single premium of a \$1,000 one-year term insurance contract is equal to the probability of the insured dying multiplied by the benefit if death occurs (\$1,000) and again multiplied by a present value factor (PVF). Using the 1980 CSO table and a 4 percent interest factor, the net single premium for a male age 30 would be calculated as follows: NSP = d / l x \$1,000 x 1/[(1 +.04) ] (Formula 3) = 16,573 / 9,579,998 x \$1,000 x = x \$1,000 x = \$ Thus, if the life insurance company collects \$ from 10,000 insureds and the premium is invested at 4 percent, approximately \$17,300 (10,000 x \$ x 1.04) will be accumulated. If exactly times the number of insureds in the group die, there will be enough assets to pay the death benefits at the end of the year ( x 10,000 x \$1,000 = \$17,300), and the insurance company would break even after all death claims have been paid. The above calculation assumes that the investment rate is 4 percent. 9 This rate of interest might seem low compared to current interest rates; but from the life insurance companies' point of view, the interest rate elected must be conservative. Because of the possible length of the life insurance contract, the debtor/creditor position of the company 8 The ability to renew varies from company to company. Some insurance companies issue term to age 99 and charge the required premiums. Many people cannot renew because they cannot afford the price. 9 Four percent is used for illustrative purposes only. Life insurance companies use varying rates in their computations. However, the rates used tend to be on the conservative (low) side.

19 toward its insureds and beneficiaries, as well as investment laws and regulations, investment of funds must be low-risk/low-return. When the interest rate is chosen for the calculation of the insurance premium, a conservative posture must be reflected. Otherwise, if high-risk investments are made and capital is lost due to a flawed investment policy, sufficient funds may not be available to pay claims as they come due. Multiple-year term contracts. To calculate the net single premium of a multi-year term insurance contract, all one must do is repeat the previous calculation (Formula 3) for the number of contract years desired. The procedure for determining the net single premium for a five-year term contract is shown in Table 4. TABLE 4 NSP Five Year Term Policy Probability of Year Age Yearly Probability of Death x Benefit x PVF = NSP ,573 / 9,579,998 1, ,023 / 9,579,998 1, ,470 / 9,579,998 1, ,200 / 9,579,998 1, ,021 / 9,579,998 1, Net Single Premium The net single premium calculation for the first year is exactly the same for the one-year term policy at age 30. In the second year, at age 31, the probability of death is calculated by taking the number of deaths predicted in the second year and dividing that number by the number of individuals starting the group at age 30. The probability required is the conditional probability of an individual dying in the second year given that the individual started at age 30. As stated earlier, the probability of an individual dying during his 31st year is not the same as the probability of death given the person started the group in an earlier year. The sum of the five probabilities shown in Table 4 equals the probability of dying during the five-year period (88,287 / 9,579,998 = ). Notice that the sum does not

20 equal one since many individuals in the group will survive. Death is not a certainty in the term contract, whereas in the permanent forms of insurance a death claim will certainly be paid if the insured does not allow the policy to lapse. The probability of surviving the five-year period is ( ). The present value factor (PVF) in Table 4 is the present value factor of \$1, given the number of years the money will be invested. The amount of money needed to pay the mortality costs for the third year is \$1.823 (\$1,000 x 17,470 / 9,579,998). If the present value of \$1.823 (\$1.621) is invested at 4 percent, then a sufficient amount of money will be available to pay death claims. Again, one may use present value tables for the factors (PVF), or one may calculate the factor using the following formula: PVF = 1 / (1 + i) n where i = Rate of interest per period n = The number of periods discounted A quick method for determining the number of discounting periods (n) in the formula is to take the age of the individual for that contract year, subtract from that number the age at issue, and add one to the result. In the above example, for the year when the insured is 34 the number of discounting periods is 5 years ( ). Calculating the net single premium of a 10-year, 15-year, or term to age 65 contract follows the same procedure, except more years of calculations are included in the final sum. Whole Life Insurance Determining the net single premium for a whole life insurance contract is similar to calculating the net single premium for the term insurance contract. Whole life insurance contracts are considered permanent insurance contracts since the insured

21 can maintain the policy throughout life. Because of the permanent nature of the contract, a calculation must be made for each year that there is a possibility of a benefit payment. Since the 1980 CSO Table assumes each insured has died by his 100th birthday, calculations need only be made through age 99. Even if this assumption is violated and the insured lives longer than age 99, the insurance company will not collect any more premiums. The life insurance company will assume the insured has died and will pay the face amount of the policy as a maturity value. The reason that the life insurance company pays the claim even if the insured is alive will become evident in Chapter 23, where policy reserves and cash values are studied. Table 5 extends the example of calculating the net single premium for the oneyear and five-year term policies for the whole life policy. The first line is exactly the same as the one-year term policy, and the first five lines are exactly the same as the five-year term policy. The calculations continue for 70 years. Only the first and last five years for the whole life calculation are shown. The net single premium for a whole life insurance policy is equal to \$209.18, using the 1980 CSO Table and a 4 percent interest rate. TABLE 5 NSP Whole Life Insurance Probability of Yearly Year Age Death x Benefit x PVF = NSP ,573 / 9,579,998 1, ,023 / 9,579,998 1, ,470 / 9,579,998 1, ,200 / 9,579,998 1, ,021 / 9,579,998 1, ,412 / 9,579,998 1, ,805 / 9,579,998 1, ,054 / 9,579,998 1, ,693 / 9,579,998 1, ,757 / 9,579,998 1,

22 Net Single Premium Endowment insurance 10 The endowment life insurance contract provides both a death and a survivor benefit. If the insured dies during the term of the contract, the face amount is paid to the beneficiary. But if the insured survives the term of the contract the face value is paid to the policyowner/insured. Thus, not only must the net single premium calculation take into account the contingency of death, but it must also take into account the possibility of the insured surviving the endowment period. The net single premium associated with the first benefit, to pay upon death, is calculated exactly like a term insurance policy (Table 6). The second benefit (Table 7), payment if the insured survives, is added to the net single premium of the first benefit to equal the net single premium of the endowment insurance policy. At the beginning of year six (end of year five), 9,491,711 individuals have survived out of the original 9,579,998 (99.1 percent). The benefit (\$1,000) and the present value factor remain unchanged from the last calculation of the first promise (to pay upon death). TABLE 6 Endowment Contract: Death Benefit Year Age Probability of Death x Benefit x PVF = Yearly NSP ,573 / 9,579,998 1, ,023 / 9,579,998 1, ,470 / 9,579,998 1, ,200 / 9,579,998 1, ,021 / 9,579,998 1, The mathematics exploring the endowment contract is provided for historical purposes since a very small proportion of life insurance sold is of this form. In 1985 less than 1/2 of 1 percent of the total life insurance in force was of the endowment type.

23 NSP to pay death Benefit TABLE 7 Endowment Contract: Survivor Benefit Year Age Probability of Death x Benefit x PVF = Yearly NSP ,491,711 / 9,579,998 1, Annuities Plus: NSP death benefit NSP for five year endowment Instead of needing contracts that pay either upon death or at the end of a specified period if the insured survives, individuals may require a contract to make periodic payments over a length of time. Annuity contracts are defined as ones requiring insurance companies to make regular payments. Often annuity payments are made to a beneficiary as one of the settlement options under the life insurance contract. Among the annuity type of settlement options provided by most life insurance companies are the fixed-payment option, the fixed-amount option, and the life income option. Other applications of annuity contracts include use in deferred compensation plans and individual retirement or group pension plans. Because the mathematical complexity increases substantially when joint life annuities are considered, this discussion will be confined to single life annuities. Joint life and group annuities are written on two or more lives, and the probabilities of survival (payment) depend on compound probabilities. Annuity certain. Not all annuities are paid based upon a life contingency. Some annuities pay rent for a period of time and terminate on a certain date regardless of

24 whether the insured lives or dies. 11 Such a contract is called an annuity certain. The net single premium of an annuity paying a stated amount each period, can be calculated by discounting the rent at an appropriate rate for the proper discounting period. Since there are no life contingencies involved, the present value of an annuity factor can be used for the calculation. As an alternative method of calculating the factor, a present value factor can be applied to each payment separately to determine the present value of the stream of payments. The calculation is shown in Table 8 for the net single premium for a male age 65 purchasing a 10-year annuity certain (\$ each year at a 4 percent discount rate). 12 The net single premium for the promise to pay \$ each year for 10 years is \$ If the insurance company earns 4 percent on its investments and pays \$100 per year for 10 years, nothing will be left after the 10th payment. Notice the first payment is discounted for one year; amounts deposited at the beginning of the year have time to earn interest. TABLE 8 Immediate Annuity Certain Yearly Year Payment x PVF = NSP \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ The term "rent" means "payment" when used in conjunction with annuities. 12 Factors such as age and sex in this example are irrelevant since payment is not contingent upon surviving.

25 Total Net Single Premium \$ or Payment x PVAF 4%, 10 periods = Net single premium \$ x = \$ Immediate versus deferred annuity. 13 The above is an example of an immediate annuity. Suppose the insured wished to deposit sufficient money in one lump sum to fund the premium five years before the annuity payments are begun by the insurer. In this case, for a deferred annuity two adjustments must be made to the calculation. First, a shift in the payment pattern occurs from the end of the year to the beginning of the year. From the insurance company's point of view, the first payment is made at the end of the fifth year or the beginning of the sixth year. Second, the premium for the annuity must be discounted for five years since the insurance company will have use of the money without having to make payments to the insured. 14 Table 9 shows the calculations for a 10-year annuity certain, deferred 5 years. TABLE 9 Deferred Annuity Certain Yearly Year Payment x PVF = NSP \$ \$ \$ \$ \$ \$ \$ Immediate annuity contracts are ones that pay rent on the next payment date after the contract is started. Deferred annuities are contracts that pay rent after skipping at least one payment date from the inception of the contract. 14 Instead of applying a deferral factor as illustrated, the present value factor for each year may be used that appropriately reflects the payment pattern. This note applies to all of the examples showing how to calculate the net single premiums for deferred annuities.

26 Discount \$ for five years: \$ x = \$ \$ \$ \$70.30 Total Net Single Premium \$ If \$ is deposited and left for five years at 4 percent, \$100 may be withdrawn at the beginning of each of the next 10 years, whereupon the principal will be exhausted. Again no life contingencies are involved in the calculation. Temporary annuities. Temporary annuities pay the insured if and only if the insured is alive; however, payments also cease once a predetermined number of payments have been made. To calculate a 10- year temporary annuity starting at age 65, the annuity certain calculation must be adjusted for the life contingency (Table 10). TABLE 10 Temporary Annuity Yearly Probability of Age Year Survival x Payment x PVF = NSP ,347,117 / 8,467, \$ ,215,925 / 8,467, \$ ,072,853 / 8,467, \$ ,917,079 / 8,467, \$ ,747,883 / 8,467, \$ ,564,669 / 8,467, \$ ,366,997 / 8,467, \$ ,154,570 / 8,467, \$ ,927,098 / 8,467, \$ ,684,331 / 8,467, \$53.37 Net Single Premium \$ According to the 1983 Annuity Table, 8,467,345 individuals are in the initial group. Of that number, 8,347,117 will collect the first rent payment since they will survive the first year. Similarly 8,215,925 members will collect the second rent payment since they will survive out of the original group. The probability of receiving a rent

27 payment for the remaining eight years is calculated in a similar fashion to the first two years. Since there is no promise to pay beyond the 10th year regardless of whether the insured is alive, no more calculations need be made. If the temporary annuity is deferred instead of immediate, the premium is determined by first changing the time-value-of-money calculation from the simple annuity to an annuity due; second, that result is discounted for the number of years of deferral. One must be careful, however, in making the adjustment. Adjustments such as those illustrated in Table 9 and Table 11 assume that if the insured dies within the accumulation period, all monies will be returned to a beneficiary or the insured's estate with a nominal amount of interest added. If the opposite is true, the amounts are forfeited to the group and the survivor rate must be calculated differently (see Table 12). The survivor rate is calculated by using the number of people who will receive the rent relative to the number of people starting the group at the beginning of the accumulation period, not at the time that the rent starts. Assume Joe Wilcox has two children. Joe decides to start saving for retirement and purchases a temporary annuity deferred for 25 years. It would be in Joe's best interest to purchase the annuity contract with the refund feature. If Joe dies before his children are on their own, the accumulated savings should go toward the children's support. On the other hand, consider the case of Fred Wheeler, who is single and desires to save for retirement. Fred has no dependents and has no plans for marriage; he would tend to choose an annuity contract without a refund feature since the premium would be less expensive and the financial security of dependents is not a consideration. Note the difference between Table 11 and Table 12. In Table 11 the number of individuals starting the group is equal to 8,577,575 at age 65. The 1983 Annuity Table predicts that 8,577,575 individuals will be alive at the beginning of age 65. On the other

28 hand, in Table 12, since there is no refund of accumulated savings, the denominator is equal to the number of individuals starting the accumulation process at age 60. TABLE 11 Ten Year Temporary Annuity Age 65 Deferred Five Years: Refund During Accumulation Yearly Probability of Age Year Survival x Payment x PVF = NSP ,577,575 / 8,577, \$ ,467,345 / 8,577, \$ ,347,117 / 8,577, \$ ,215,925 / 8,577, \$ ,072,853 / 8,577, \$ ,917,079 / 8,577, \$ ,747,883 / 8,577, \$ ,564,669 / 8,577, \$ ,366,997 / 8,577, \$ ,154,570 / 8,577, \$58.64 \$ Deferral Factor X Net Single Premium \$646.29

29 TABLE 12 Ten Year Temporary Annuity Age 65 Deferred Five Years: No Refund During Accumulation Yearly Probability of Age Year Survival x Payment x PVF = NSP ,577,575 / 9,013, \$ ,467,345 / 9,013, \$ ,347,117 / 9,013, \$ ,215,925 / 9,013, \$ ,072,853 / 9,013, \$ ,917,079 / 9,013, \$ ,747,883 / 9,013, \$ ,564,669 / 9,013, \$ ,366,997 / 9,013, \$ ,154,570 / 9,013, \$55.80 \$ Deferral Factor x Net Single Premium \$ Whole life annuities. The whole life annuity contract pays rent until the insured dies. To determine the net single premium for a whole life annuity, one need only extend the number of calculations in the above temporary annuity calculation by adding the remaining discounted survival benefits. For brevity, the first and the last three years of the whole life immediate annuity are shown in Table 13. The 1983 Annuity Table predicts that there will be no survivors in the 116th year; thus calculations cease in the 115th year. The same methods used to modify the immediate temporary annuity to a deferred temporary annuity are used to convert the whole life annuity from immediate to deferred. First, one must recognize that the payments are made at the beginning of each period (annuity due) instead of the end of the period. Second, because of the shift in the timing of the payment the time-value-of-money factors must also change accordingly. Again, this modification assumes that during the accumulation period assets held by the insurance company are refunded upon the death of the insured with a nominal amount of interest added. If there is no refund during accumulation, as

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