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NET SINGLE PREMIUM

The first step in deriving any premium is to find the net single premium for that policy. Then a set of more frequent premiums that are equivalent to the single premium may be developed if needed, as is usually the case. Finally, an amount is added for expenses and contingencies. This section describes how to calculate the net single premium.

An important feature of premium calculation develops because mortality tables display annual rates. Calculations start with the assumption that premiums are paid at the beginning of each policy year and benefits are paid at the end of the policy year. Annuity contract calculations also start with payments made annually. Then premiums for contracts with benefits and payments more than once yearly are computed by assuming that deaths occur uniformly throughout the policy year.

Concept of the Net Single Premium

The objective of life insurance rate making is to assure that the company collects enough from each group of insureds to pay the benefits promised under the contract. If the contract is purchased with a single or lump sum, that sum is the present value of future benefits. Rate making, the process of valuing the promises in the contract, involves three steps.

The first step is to learn the benefits promised under the contract and the length of time that the promise remains in effect. Life insurance benefits take two basic forms: (1) a death benefit that is the company�s promise to pay a specified amount�called the face amount�if the insured dies while he or she is covered and (2) an endowment benefit that is the company�s promise to pay a specified amount if the insured should survive to the end of the covered period. Some contracts contain both promises, while others contain only the first.

The second step is to select a mortality table to use in measuring the probabilities involved. Whether the company�s promise is to pay upon death, survival, or both, the rate of mortality determines the value of the promise. Unless otherwise stated, this book uses the 1980 Commissioners Standard Ordinary (CSO) Mortality Tables to illustrate premium computations. In practice, however, a company must adopt the most appropriate mortality table for the group of persons insured.

The third step is to select an interest rate to be used in adjusting for the time value of money. The fact that premiums are paid in the present while the benefits to be received from the company must be fulfilled in the future significantly reduces the cost of all forms of insurance. (See chapter 14 for discussions of the time value of money.) The rate at which expected benefits are discounted greatly influences the size of the net single premium. The lower the rate, the higher the premium, and vice versa. All calculations in this section assume 5.5 percent interest.

Two Techniques of Calculation

The net single premium is always the sum of the present values of all the expected benefits. It may, however, be computed according to either of two techniques. The first uses probabilities for each insured and is called the individual approach. The second assumes a large group of insureds and is called the aggregate approach.

The individual approach incorporates the necessary three steps by finding the product of these three factors for each year of the contract:

 

(1) the amount (which is defined in the contract)

(2) the discount factor (as discussed in chapter 14)

(3) the probability of payment (as shown in the mortality table)

 

The probability of dying, factor (3), is a scientific estimate that will always contain an element of uncertainty. That uncertainty is acceptable, however, if a company sells a sufficiently large number of policies. As discussed in chapter 13, the law of large numbers assures that the average of many such payments actually made will be near the value expected�that is, (1) x (2) x (3).

The individual approaches can be illustrated with the calculation of the net single premium for a one-year term policy issued to a male at age 32. This contract pays the $1,000 face of the policy if the insured dies during the year of coverage and nothing if he survives. The three factors are as follows:

 

(1) the possible amount of $1,000

(2) the discount factor 1/$1.055 = $0.947867 that reflects the fact that any payment will be made at the end of one year

(3) the probability that a 32-year-old male will die within one year. According to the 1980 male CSO Table, 17,470 out of 9,546,404
males aged 32 will die within the year. Thus the probability is 17,470/9,546,404 = .001830.

 

The expected present value of this uncertain payment is

 

($1,000) x (.94787) x (.001830) = $1.73

 

The aggregate approach, on the other hand, uses the large numbers shown in a mortality table directly and requires less proficiency with combining decimal numbers. The process begins by assuming that the number of persons shown in the mortality table as living at the issue age is the starting population for the insurance arrangement. The rates in the table predict how many payments are expected to occur at each subsequent age. A discount factor reduces the aggregate payments at each age to the present value. In the final step, the total of these present values is divided by the number of lives at issue age.

The two methods give identical premiums. This is demonstrated by recalculating, using the aggregate method, the net single premium for the same policy as was shown earlier. To apply the 1980 male CSO Table using the aggregate approach, 9,546,404 males aged 32 are assumed to apply for a one-year term policy for $1,000. Of those, 17,470 die during the following year. Therefore the company must have $17,470,000 on hand at the end of the year to pay claims. Some of that money can come from interest earned during the policy�s one-year term. The insurance company needs to collect only the present value of this amount from the policyowners at issue. Therefore, the amount that must be collected from the group is $17,470,000 divided by $1.055, or $16,559,242. Since it is not known at the beginning of the year which persons will die, each must pay the same amount into the fund. That is the total contribution divided by the 9,546,404 receiving coverage:

 

$16,559,242 � 9,546,404 = $1.73

Since both techniques have educational value, their use will be alternated in this chapter�s illustrations.

Term Insurance

Term insurance policies that provide protection on a level premium basis for several years are important in practice and for illustration. The net single premium for a 5-year term policy for $1,000 issued to a female aged 32 will be calculated by the individual approach.

The individual approach was defined earlier for one uncertain future payment. The 5-year term policy has five uncertain future payments. The present value of each expected future payment is calculated. The net single premium for the term insurance is the sum of these expected present values. Stated differently, the net single premium, developed using the individual approach, is the sum of the following five products:

 

(1) For the possible payment at the end of the first policy year, the amount is $1,000, the discount factor for one year is 1/1.055, and the probability of payment is, according to the 1980 female CSO Table, 14,037 out of 9,680,912, or 0.00145. The expected present value is the product of these factors:

 

$1,000 x 0.94787 x 0.00145 = $1.37

 

(2) For the possible payment at the end of the second policy year, the amount is $1,000, the discount factor for 2 years is (1/1.055)2, and the probability of payment is, according to the 1980 female CSO Table, 14,500 out of 9,680,912, or 0.00149. The expected present value is the product of these factors:

 

$1,000 x 0.898452 x 0.00149 = $1.35

 

(3) For the possible payment at the end of the third policy year, the amount is $1,000, the discount factor for one year is 1/(1.055)3, and the probability of payment is, according to the 1980 female CSO Table, 15,251 out of 9,680,912 or 0.00157. The expected present value is the product of these factors:

 

$1,000 x 0.851614 x 0.00157 = $1.34

 

(4) For the possible payment at the end of the fourth policy year, the expected present value is

 

 

(5) For the fifth policy year, the expected present value is

 

 

In summary, the net single premium for the 5 years of coverage is

 

1.37 + 1.35 + 1.34 + 1.33 + 1.34 = $6.73.

 

The aggregate approach achieves this same result by adding the terms before dividing by the number of policyowners who are alive to share the costs at issue, 9,680,912.

Whole Life Insurance

An insurance policy that provides protection for the whole of life is called whole life insurance. The face amount of the policy is payable upon the insured�s death, whatever the insured�s age. Eventual payment of the face is certain. The only uncertainty is the year in which the policy will become a claim. However, because whole life also can be viewed as a term insurance policy for the remaining life span, the techniques for computing the net single premium are the same as for a term policy.

Under the individual approach, the net single premium for a whole life insurance policy issued to a male aged 32 is the sum of the expected payments at the end of his 33d year, his 34th year, and every year up to and including his "last age" according to the table being used for the calculation. Since this is age 100 in the 1980 CSO Tables, the example involves 68 separate probabilities. For the sake of brevity, table 15-1 shows only the equations for the first 5 and last 5 years.

Adding the results of these 68 computations produces a sum of $140.23. That number represents the expected present value of possible payments at the end of each year from age 32 to 99, inclusive.

One interesting and significant result arises from the fact that a whole life policy inevitably will become a claim. The net single premium at any age of issue would be $1,000 per $1,000 of insurance except for the interest earnings on the advance deposit. The entire process of calculating the probability that death will occur at each of the possible ages is important only to help find the amount of interest that will be earned on the advance premium before the death claim must be paid.

Endowment Insurance

An endowment insurance contract promises to pay a death benefit if the insured should die during the term of the contract or to pay an endowment amount (usually equal to the death benefit) if the insured should survive to the end of the term. Endowment contracts have been dropped by United States life insurers as a result of federal income tax law changes. These policies do not satisfy the tax code definition of life insurance if they endow at ages below 95. Endowment policies that were issued before 1985 and have been kept in force continue to be treated as life insurance under the transition provisions of the tax

 

TABLE 15-1
Calculating Net Single Premium for Whole Life Insurance Policy
Issued to Male Aged 32

Age 32: Cost of first year�s mortality

 

Age 33: Cost of 2d year�s mortality

 

Age 34: Cost of 3d year�s mortality

 

Age 35: Cost of 4th year�s mortality

 

Age 36: Cost of 5th year�s mortality

 

· · ·

 

Age 95: Cost of 64th year�s mortality

 

Age 96: Cost of 65th year�s mortality

 

Age 97: Cost of 66th year�s mortality

 

Age 98: Cost of 67th year�s mortality

 

Age 99: Cost of 68th year�s mortality

code. Endowment policies are still available in many countries. They are even the most frequently purchased type of life insurance in some countries where high savings rates are common.

An endowment insurance contract is a combination of a pure endowment and term insurance. The former pays only if the insured survives the specified period of years, and the latter pays only if the insured does not survive the specified period. The net single premium for the endowment insurance contract is then the sum of the respective net single premiums for the pure endowment and term insurance contracts.

The net single premium for a $1,000 10-year endowment contract issued to a female at aged 45 is the sum of these same two components. Using the aggregate approach we calculate the amount that, if invested at issue in a 5.5 percent fund, would be sufficient to pay the term insurance benefits of the contract for 9,409,244 45-year-old females (9,409,244 is the number living at age 45 in the 1980 female CSO Table). The calculation is shown in table 15-2.

 

 

TABLE 15-2
Calculating Net Single Premium for 10-year Endowment
Insurance Contract Issued to Female Aged 45

Policy Year

Benefit Amount

Number of Deaths

Discount Factor

Amount Required
at Issue

1st

$1,000

33,497 

0.947867

$ 31,750,700.90

2d

1,000

35,628 

0.898452

32,010,047.86

3d

1,000

37,827 

0.851614

32,214,002.78

4th

1,000

40,279 

0.807217

32,513,893.54

5th

1,000

42,883 

0.765134

32,811,241.32

6th

1,000

45,727 

0.725246

33,163,323.84

7th

1,000

48,711 

0.687437

33,485,743.71

8th

1,000

52,011 

0.651599

33,890,315.59

9th

1,000

55,797 

0.617629

34,461,845.31

10th

1,000

59,602 

0.585431

34,892,858.46

TOTAL

$331,193,973.31

 

 

The total required at age 45 for the 9,409,244 lives is $331,193,973.31. The net single premium is the amount per individual:

 

 

 

$331,193,973.31 + 9,409,244 = $35.20.

 

Since the pure endowment contract has only one possible payment, its net single premium can be calculated by the three-factor product that was used above. Here the factors are as follows:

 

(1) the amount $1,000

(2) the discount factor 1/(1.055)10 = 0.585431

(3) the probability of payment�the ratio of the number surviving at age 55, 8,957,282, divided by the number at issue, 9,409,244

 

Thus by the individual method, the net single premium for the pure endowment is $557.31. The net single premium for the 10-year endowment insurance contract is obtained by adding the net single premiums for the 10-year term insurance and the 10-year pure endowment:

 

Net single premium, 10-year term policy, at age 45 $ 35.20

Net single premium, 10-year pure endowment, at age 45 557.31

Net single premium, 10-year endowment insurance contract $592.51

 

It is interesting to consider the sources of funds to pay the pure endowment benefit. The net premium of $557.31 deposited with the company at the beginning of the term is available. Interest on this amount for the 10-year period will be $557.31 x (1.055)10 � 557.31, or $394.66. At the end of the 10-year period the net premium plus interest is $951.97. Yet the policy promises $1,000 if the insured is alive on that date. From what source does the difference $1,000 � $951.97, or $48.03, come?

Benefit of Survivorship

The difference is attributable to the benefit of survivorship�the pro rata share of each survivor in a fund created by the premiums, plus accumulated interest of those persons who failed to survive the period. According to the 1980 female CSO Table, 451,962 persons out of 9,409,244 alive at age 45 will die within the next 10 years. Each of those persons deposits $557.31 with the company at the inception of the contract. That money is not returnable in case of death. By the end of the period, each deposit has accumulated to $951.97, creating an aggregate fund of $951.97 x 451,962, or $430,254,265.10, which is forfeited by those who died and available to the survivors. Divided by the number of survivors, 8,957,282, the fund yields $48.03 per survivor. This sum added to the accumulation of $951.97 provides the $1,000 promised under the contract.

This concept underlies all contracts that provide benefits based on survival to the end of a specified period. The longer the period and the higher the rate of mortality, the greater the benefit of survivorship. For purposes of comparison recognize that the benefit of survivorship under a 20-year endowment issued at age 45 is $135.53 per survivor.

The following demonstrates how the entire premium for an endowment insurance contract can be derived by the group or aggregate approach:

 

Total aggregate mortality claims = 331,193,973.31 (see table 15-2)

 

Total pure endowment payments = number of lives at age 55 x 1000 x discount

factor

 

= 8,957,282 x 1000 x 0.585431 =
5,243,870,600

 

Net single premium =

 

= $592.51

Life Annuities

Chapters 6 and 14 discussed annuities certain. Certain means that payments occur unconditionally for a known term without regard to any contingencies. This chapter discusses life annuities. The adjective life defines the payments as occurring only upon survival of a designated life. The terminology used to describe life annuities is consistent with the terminology used for annuities certain. A life annuity due has its first payment due at issue or contract date; a life annuity immediate has its first payment due at the end of one payment interval; a deferred life annuity has its first payment due after the completion of a deferment period.

Life annuities play dual roles for any insurance company. The company can be on the paying end or on the receiving end of a life annuity. The company pays annuities to annuitants, beneficiaries, and retirees; these are usually annuities immediate. It receives premiums from policyowners in the form of annuities; these are usually annuities due. As we calculate the net single premiums for life annuities, remember that these concepts are applied to value both benefit payments and premium receipts.

The appropriate mortality rates differ between computations for annuities and for insurance policies. This important difference occurs simply because annuitants live longer. The table must be chosen to reflect the experience of annuitants or insured lives.

Calculating Life Annuity Present Values

To compute the net single premium for a life annuity, view the annuity as a series of pure endowments. Just as the pure endowment pays only if the insured survives to the end of a specified period, a life annuity pays only if the annuitant is alive on the date the payment is due.

Calculating the present value of the life annuity requires three dates: (1) the contract or issue date, (2) the due date of the first payment, and (3) the due date of the last possible payment. Consider each payment a pure endowment payable on its due date. The net single premium of the annuity is the sum of the net single premiums on the contract date for the payments or "pure endowments" specified by the annuity.

The dates for the net single premium of a life annuity immediate payable to a 70-year old male are as follows: (1) The contract date is at age 70, (2) the first payment is at age 71, and (3) the last possible payment is at the end of the mortality table. When using the 1983 Individual Annuity Mortality (IAM) Table for Males, the payments could extend to age 115. While the table ends at age 115, the payments would, of course, continue after age 115 if the annuitant survives. The following discussion uses the individual method to illustrate this calculation for annual payments of $100.

The probability that the first payment of $100 will be made is 7,747,886 ÷ 7,917,081 = 0.97863. The numerator is the number of persons alive at age 71; the denominator is the number alive at age 70. Since one year will elapse before the payment, if it occurs at all, the sum set aside at the time of purchase can be discounted at 5.5 percent. The present value of the first payment, therefore, is figured as follows:

 

The probability that the second payment occurs is the probability that a person now aged 70 will be alive at age 72. Again discount the contingent payment, this time for 2 years. The present value of the second payment is determined as shown in the following equation:

 

x $100 x .898452 = $85.85

 

The denominator in the first term does not change in these equations or any of the other 43 separate equations. The first five and last four equations needed to compute the net single premium for the entire series of contingent payments are shown in table 15-3.

The present value of all payments is $901.82; the present value at age 70 of a payment at extreme ages, such as 111, is zero when rounded to the nearest

 

TABLE 15-3
Calculating Net Single Premium for Life Annuity Immediate Payable to Male Aged 70
Age 71: x 100 x .947867 = $92.76 Present value of first annuity payment

 

Age 72: x 100 x .898452 = $85.85 Present value of second annuity payment

 

Age 73: x 100 x .851614 = $79.24 Present value of third annuity payment

 

Age 74: x 100 x .807217 = $72.95 Present value of fourth annuity payment

 

Age 75: x 100 x .765134 = $66.95 Present value of fifth annuity payment

 

· · ·

 

Age 111:x 100 x .11134 = $.000685 Present value of 41st annuity payment

 

Age 112: x 100 x .105535 = $.000197 Present value of 42d annuity payment

 

Age 113: x 100 x .100033 = $0.000044 Present value of 43d annuity payment

 

Age 114: x 100 x .094818 = $0.000007 Present value of 44th annuity payment

 

 

 

cent. We show the figures to six decimals to emphasize that only the last computation produces an answer that is literally zero.

Thus in consideration of $901.82 paid in a single sum at the inception of the contract, an insurance company could afford to pay a 70-year-old man an income of $100 per year as long as he lives, the first payment being made at age 71. The computation of the net single premium for such an annuity can also be viewed on an aggregate basis. Either premium computation presumes that the company enters into sufficient contracts to experience average results.

The net single premium for an immediate annuity with guaranteed payments for a specified number of years, whether the annuitant survives or not, is similar to the immediate annuity above. Simply replace the probability of payment with 1.00 for each year during the certain period. Because no contingency is involved, the discount factor is the only cost-reducing factor during this period.

To take a simple example, consider an immediate annuity purchased at age 70 to provide an income of $100 per year with the payments guaranteed for 5 years. Using the 1983 IAM Table for Females and the aggregate method, 8,837,346 payments are assumed to be payable at the end of each of the first 5 years instead of the number of persons shown as living in the mortality table at those ages. The mortality table values are used as before beginning with the sixth payment. Determining the net single premium for the entire series of payments involves 45 separate equations. The first 10 and last five are shown in table 15-4. The sum of the present values of these 45 payments is $1,045.88.

The net single premium for a temporary life annuity is calculated using the same underlying principles as the calculations in table 15-4. Since the promised payment is zero after the term of the annuity, the computations end with that age. For a 10-year life annuity issued to a male at age 70, for instance, the first probability would be the chance of survival to age 71, or

 

= 0.97863

 

and the last probability would be the chance that the annuitant would survive to age 80, or

 

= .70229

Deferred Whole Life Annuity

The amount that must be on hand at age 70 to provide a life income of $100 per year, with no payments guaranteed, was shown in table 15-3 to be $901.82. This amount may be paid to the insurance company in a single sum at the purchaser�s age 70. Alternatively, the present value of that amount may be deposited with the company years before the time the income is to commence. More likely still, the present value may be accumulated through a series of periodic deposits before the income is to begin. If the funds are deposited with the company before annuity payments begin, a smaller premium is required.

The adjustment can be most clearly explained in terms of a nonrefund annuity purchased with a single premium some years before the annuity starting date. Assume that a male aged 30 purchases an annuity contract that will pay him an annual income of $100 for life beginning at age 70. Under the contract�s terms nothing is paid or refunded in the event of his death before age 70. Note that in this example the income is to begin one year earlier than the earlier example that derived the premium for an immediate annuity.

 

TABLE 15-4
Calculating Net Single Premium for Immediate Annuity with
Payments Guaranteed for 5 Years for Female Aged 70

Age 71: 1  x 100 x .947867 = $94.76 Present value of first year�s payment

 

Age 72: 1  x 100 x .898452 = $89.85 Present value of 2d year�s payment

 

Age 73: 1  x 100 x .851614 = $85.16 Present value of 3d year�s payment

 

Age 74: 1  x 100 x .807217 = $80.72 Present value of 4th year�s payment

 

Age 75: 1  x 100 x .765134 = $76.51 Present value of 5th year�s payment

 

Present value of 5-year annuity certain � $427.03

Age 76: x 100 x .725246 = $66.04 Present value of 6th year�s payment

 

Age 77: x 100 x .6874368 = $61.18 Present value of 7th year�s payment

 

Age 78: x 100 x .65159887 = $56.51 Present value of 8th year�s payment

 

Age 79: x 100 x .617629 = $52.02 Present value of 9th year�s payment

 

Age 80: x 100 x .585431 = $47.72 Present value of 10th year�s payment

· · ·

 

Age 111: x 100 x .11134 = $0.00 Present value of 41st year�s payment

 

Age 112: x 100 x .105535 = $0.00 Present value of 42d year�s payment

 

Age 113: x 100 x .100033 = $0.00 Present value of 43d year�s payment

 

Age 114: x 100 x .094818 = $0.00 Present value of 44th year�s payment

 

Age 115: x 100 x .089875 = $0.00 Present value of 45th year�s payment

Pure Endowment and General Annuity Approaches

The premium for this deferred annuity can be calculated in two different ways. First is the "pure endowment approach." It consists of calculating the net single premium for an immediate life annuity providing $100 per year, with the first payment at 70, and considering this net single premium to be the amount of a pure endowment promised at age 70. Next the net single premium at age 30 for this pure endowment due at age 70 is calculated.

To illustrate the pure endowment approach recall that $901.82 for a male at age 70 will provide a life income of $100 per year beginning at age 71. With $100 added to this sum, the $100 payments could begin at age 70, the additional $100 taking care of the payment to be made on the effective date of the contract. Since no time elapses, no interest is earned and no life contingency is involved. Therefore the net single premium at age 70 for an annuity that will provide $100 immediately and $100 per year thereafter as long as the annuitant lives is $1,001.82.

A sum less than $1,001.82 deposited with the company at age 30 will provide that same payment stream commencing at age 70. Two reasons justify a substantial reduction: First, funds deposited at age 30 will earn interest for 40 years. Second, a substantial probability exists that the purchaser will not survive to age 70 to receive payments. The sums forfeited by those who fail to survive to that age reduce, through the benefit of survivorship, the amount that each annuitant must pay at the outset. Therefore the sum that must be deposited with the company at age 30 is as follows:

 

x $1,001.82 x 0.117463 = $94.31 (the net single premium)

 

The second method available to determine the net single premium for an annuity is the "general annuity" method. Using the same example as above, each payment is treated as a separate pure endowment to be discounted to age 30. The first payment of $100 will be discounted for 40 years and multiplied by the probability of the annuitant�s surviving to age 70. The second payment will be discounted for 41 years and multiplied by the probability of survival to 71, and so on to age 114 when the 1983 IAM Table is used. The first five and last five equations of the process are set forth in table 15-5 using the 1983 IAM for females and 5.5 percent interest.

The sum of the present values of all 45 payments is $117.96. This compares with a value of $94.31 developed earlier for males. The difference illustrates vividly the spread between male and female mortality.

The net single premium can also be obtained by using the aggregate approach as shown in table 15-6. As usual, only the first five and last five equations appear in the table.

 

 

TABLE 15-5
Calculating Net Single Premium for Deferred Whole Life
Annuity for Female Aged 30 (General Annuity Method)

 

 

· · ·

 

 

 

 

 

 

 

 

 

 

 

TABLE 15-6
Calculating Net Single Premium for Deferred Whole Life Annuity for Female Aged 30 (Aggregate Approach)

1st payment: 8,837,346 x 100 x 0.117463 = 103,806,117.30

2d payment: 8,733.975 x 100 x 0.111339 = 97,243,204.25

3d payment: 8,621,263 x 100 x 0.105535 = 90,984,499.07

4th payment: 8,497,815 x 100 x 0.100033 = 85,006,192.79

5th payment: 8,362,020 x 100 x 0.094818 = 79,287,001.24

 

· · ·

 

 

41st payment: 2,640 x 100 x 0.013078 =

42d payment: 922 x 100 x 0.012396 = 3,452.59

43d payment: 254 x 100 x 0.011750 = 1,142.91

44th payment: 40 x 100 x 0.011138 = 298.45

45th payment: 5 x 100 x 0,010557 = 54.28

              5.28

Present Value of payments = 1,172,546,028.40

 

Net single premium = 1,172,546,028.4 = $117.96

9,939,983

 

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