Few life insurance contracts are purchased with single premiums. Few persons have sufficient savings to buy adequate life insurance on a single-premium basis. This would also run counter to the prevailing practice in consumer finance, where installment purchases have become the pattern.

Apart from the trends of the times, financing life insurance on an installment basis is appropriate. Since the fundamental purpose of life insurance is to provide protection against the loss of future earnings�which by definition are received in periodic installments�paying the cost of that protection over a similar time frame is logical.

Another reason most people prefer installment financing of life insurance is the lower total cost if the insured dies early. One monthly premium purchases as much life insurance protection as a single premium�only the period of coverage is reduced. If the insured dies within a few years after the single-sum purchase of a life insurance policy, the cost will be many times greater than if annual (or more frequent) installment payments are made. On the other hand, if policyowners live beyond the period, the total amount of the annual premiums paid will exceed the single premium, and with each passing year the disparity will become greater.

Policyowners must be given a fair choice between paying for insurance with a single premium and with a set of level annual premiums. The prices must be determined in a manner that leaves the financial position of the company unaffected by the policyowner�s decision. Such pricing is referred to as actuarially equivalent pricing. To be the actuarial equivalent of a policy�s net single premium, its net level premiums must reflect (1) the possibility that the insured may die having made only some of the payments and (2) the smaller amount invested that will reduce the investment earnings to the company. Expressed in positive terms the net level premium must reflect (1) the probability that the insured will survive to pay premiums and (2) the period during which the premiums will earn investment income.

Deriving the net level annual premiums integrates two computations described earlier in this chapter. No new computational skills are required. The rule for determining net level annual premiums is this: divide the net single premium for the policy in question by the present value of a life annuity due of $1 for the premium-paying period. The process is illustrated on the following pages using net single premiums derived above.

The earlier example found the net single premium for a $1,000 5-year term policy issued to a female aged 32 to be $6.73. What level annual premium paid at the beginning of the contract and on each of the next four anniversary dates, if the insured is then living, is the equivalent of $6.73? The answer to this question, following the rule stated above, requires that one know the present value of a temporary life annuity due of $1 for a term of 5 years for a female aged 32. The 1980 CSO Table is used for this calculation since that is the table used to derive the $6.73 premium. In other words, the same mortality and interest assumptions must be used to determine the present value of the annual premiums as are used to determine the present value of benefits. Since the mortality rates of the 1980 CSO Tables are greater than the mortality rates of the 1983 Individual Annuity Tables, the present value of an annuity due calculated according to the 1980 CSO Table is smaller than the present value of an annuity due based on the 1983 Individual Annuity Table. The present value is computed as shown in table 15-7.

The sum of the five payments, the present value of a 5-year temporary annuity due of $1 for a female aged 32, is $4.50. This means that a premium of $1 for each period will purchase a policy with a net single premium of $4.50. The net single premium for the 5-year term policy, however, is $6.73. Hence, to determine the size of five level annual premiums payable beginning at age 32 that are equivalent to $6.73, divide $6.73 by $4.50. The net level annual premium for a $1,000 5-year term policy issued to a female aged 32 is $6.73 ÷ $4.50 = $1.50. The maximum amount any policyowner might pay is $7.50�five annual premiums of $1.50 each. This exceeds the net single premium of $6.73.

Ordinary life insurance is whole life insurance for which premiums will be paid throughout the lifetime of the insured. To obtain the net level annual premium for an ordinary life policy, the net single premium for a whole life policy is divided by the present value of a whole life annuity due of $1. Since the whole of life is the longest premium-paying period contemplated under any whole life insurance policy, the present value per $1 per annum is greater than for any other premium-paying period. This produces the lowest level annual premium of any whole life policy because the net single premium for a whole life policy at any particular age is the same, regardless of the premium paying-period. The longer the period over which the premiums are spread, the smaller each periodic premium will be.

Earlier in this chapter, the net single premium for a whole life insurance for a 32-year-old male was determined. Here the present value of a life annuity due of $1 for the whole of life for a male aged 32 is calculated by the aggregate approach. Then the level annual premium for an ordinary life insurance policy can be determined.

The present value of the life annuity due must recognize the possibility that the insured may not be alive at age 33 (and each age thereafter to the end of the mortality table) to pay the second and subsequent premiums. The probability of survival is the probability of a payment occurring. The first five and last five computations are shown in table 15-8.

As the last line of table 15-8 shows, the present value of the 68th payment is very slight, only $0.00003. Still, it must be taken into account. The sum of all the payments� present values is $16.49. Dividing this number into the net single premium of $140.23 for a whole life policy issued at age 32 produces a net level annual premium of $8.50. Stated differently, the $1,000 whole life policy that costs $140.23 if purchased with a single-sum payment can also be purchased with premium payments of $8.50 annually.

The net single premium for a whole life policy can be spread over any number of years by means of the appropriate life annuity due. Suppose a male policyowner, aged 32, wants to pay for his policy in 20 annual installments. Then the present value of a 20-year temporary life annuity due of $1 per annum is determined. This annuity�s first payment is at age 32 and its last possible payment is at age 51. The present value of such an annuity is $12.35. This number is smaller than the corresponding whole life annuity due and divided into the same net single premium, $140.23, produces a larger level annual premium. The level annual premium for a 20-payment life policy issued at age 32 is

$140.23 + $12.35 = $11.36.

For a 10-payment life policy, end the computations at age 41. Following the formula previously given, this yields a present value for the temporary life annuity due of $7.88. Dividing $7.88 into $140.23 gives a 10-payment life premium of $17.80.

Finally, if the whole life policy is to be paid up at age 65, the annuity value needed for the denominator is the present value of a series of payments of $1 per year, extending from age 32 to 64 and contingent on the insured�s survival.

The net level premium for an endowment insurance policy is derived in exactly the same manner as that of any other policy. Once again, the procedure to determine the net level annual premium is to divide the net single premium by

the present value of a temporary life annuity due of $1 for the premium-paying period. All endowment policies, of course, are limited-payment policies in the sense that the premium is not payable for the whole of life. Usually premiums are payable for the full term of an endowment insurance contract.

As found earlier in this chapter, the net single premium for a $1,000 10-year endowment issued to a female aged 45 is $592.51. The present value of a 10-year temporary life annuity due of $1 for a female at age 45 is $7.82. The net level annual premium, therefore, is $592.51 ÷ $7.82, or $75.77.

Deferred annuities usually are financed with annual�rather than single�premiums. Premiums may be paid throughout the period of deferment or may be limited to a shorter period of years. The annuity contract may promise to return the annuitant�s premiums with or without interest. The annual premium in this case does not involve life contingencies but rather is a sum of money that must be set aside annually to accumulate at an assumed rate of compound interest to a predetermined amount at a specified date. For example, $1,151.88 must be on hand at age 65 to provide an income of $100 per year to a male. How much would a male aged 40 have to set aside each year, including a payment on his 40th birthday, to accumulate a sum of $1,151.88 by his 65th birthday, assuming such annual payments earn compound interest at the rate of 5.5 percent? Such a program includes 25 payments, the first at age 40 and the last at age 64. The period of accumulation is 25 years. At the end of 25 years $1 per year earning 5.5 percent interest will accumulate to $53.9660. Dividing $1,151.88 by $53.9660 shows that $21.34 must be set aside during each of 25 years to accumulate the required single premium. This computation presumes that premiums are to be returned if the annuitant dies before age 65.

Alternatively, the contract may provide that the company retains all premiums paid in the event of death before the annuity income commences. If premiums will not be refunded, the net level annual premium for a deferred annuity is determined by the same methods as life insurance. Again, however, the premium annuity�s present value must be computed using the same mortality rates as those used for the benefit annuity. Using the 1983 Individual Annuity Mortality Table, the net level premium for a nonrefund deferred annuity purchased by a female aged 40, with income to begin at 65, is computed as a temporary life annuity due. That annuity makes one payment immediately with 24 subsequent payments due. The net single premium for that annuity is $311.38. The net level premium is $311.38 ÷ $13.95198, or $22.32.