As stated earlier, reserves may be ascertained prospectively or retrospectively. Since the retrospective method is easier to grasp, it will be discussed first.

The retrospective method of valuation can be explained in terms of either one policy or all policies in a given classification. The group or aggregate approach will be described first.

The retrospective reserve arises out of the level premium design. Under that arrangement premiums in the early policy years are more than adequate to cover the death claims, thereby creating a fund that can be drawn upon in the later years of coverage when death rates rise sharply and claims exceed current premium income. In that sense the reserve may be regarded as an accumulation of unearned premiums (a form of partial-prefunding somewhat similar to a layaway plan for merchandise purchases). The retrospective reserve is sometimes even described as the unearned premium reserve. The prefunding generates investment income that benefits the surviving policyowners, showing up on the company�s financial statement as a liability item. As a matter of fact, reserves in the aggregate constitute the major liability item on the insurer�s balance sheet, typically accounting for about 90 percent of all insurer liabilities. Most of an insurance company�s assets (typically 80 percent) are held in offset to policy reserves and are for the benefit and protection of the policyowners. Premiums are lower than they would have to be if there were no investment earnings associated with reserves. The exact manner in which prefunding leads to the creation of reserves is demonstrated in table 16-1 (which is shown at the end of this chapter).

Table 16-1 (at the end of this chapter) shows the progression of reserve funds on a group of ordinary life contracts issued to males aged 35 and written in the amount of $1,000. According to the 1980 CSO Male Table, 9,491,711 persons would be alive at age 35 out of an original group of 10 million individuals alive at age zero. Therefore it is assumed in the illustration that the group of people taking out an ordinary life contract at age 35 is composed of the survivors of the original group of 10 million. The net level premium for an ordinary life contract issued at age 35, computed on the basis of the 1980 CSO Male Table and 4.5 percent interest, is $11.60 per $1,000 of face amount. Thus the group will contribute a total of $110,144,947 (9,491,711 x $11.60433) in premiums at the beginning of the first policy year. Since it is assumed that no death claims will be paid until the end of the year, the entire sum of $110,144,947 will earn interest throughout the year at the rate of 4.5 percent, producing earnings of $4,956,523. Therefore a sum of $115,101,469 will have accumulated by the end of the year, from which anticipated death claims in the amount of $20,028,000 will be deducted. This will leave a fund of $95,073,469 that, divided equally among the 9,471,683 survivors, will yield an individual reserve of $10.04.

The sum of $95,073,469 is carried over to the beginning of the 2d policy year and is augmented by the second annual premium of $109,912,535 (9,471,683 x $11.60433), producing a sum of $204,986,004, which will likewise be invested at 4.5 percent interest throughout the year. Interest earnings of $9,224,370 will bring the accumulated sum up to $214,210,375. According to the 1980 CSO Male Table, 21,217 persons will die during the year, and the payment of their claims at the end of the year will reduce the fund to $192,993,375, or $20.42 per surviving policyowner. The terminal reserve for the 2d policy year will be carried forward to the beginning of the 3d policy year, and the process of adding annual premiums, crediting interest on the combined sum, and subtracting death claims is repeated. The process continues until the last of the policyowners is assumed to have died.

Note that in the early years net premium income greatly exceeds the tabular death claims. During that period therefore both premium and investment income contribute to building up the reserve fund. For each year that goes by, however, premium income declines (because of the reduction in the number of policyowners); while for a long period the dollar value of death claims increases. By the 23d policy year, death claims catch up with premium income (death claims exceed premium income from then on until the end of the mortality table), but for a few years thereafter, interest on the accumulated fund (including current premiums) is more than adequate to absorb the deficiency in net premium income, and the aggregate fund continues to grow. Beginning in the 35th policy year, however, death claims exceed both premium income and interest earnings, and the total fund begins to decline. In other words, after the 35th policy year, a portion of the principal must be used to pay death claims. Death claims reach a maximum in the 44th policy year, totaling $329,936,000 but, even after tapering off, continue to exceed current premiums and investment earnings. The reserve fund continues to shrink until, with the payment of the death claims of the last 10,757 survivors in the 65th policy year, it is completely exhausted.

A sharp distinction must be drawn between the aggregate reserve and the pro rata portion of that reserve allocable to an individual policyowner. As table 16-1 shows, the individual reserve increases each year, eventually equaling the face amount of the policy (before payment of the last set of death claims). It continues to increase after the aggregate reserve begins its decline because the number of survivors decreases at a faster rate than the aggregate fund. The individual reserve must accumulate to the face amount of the policy; if each group of policyowners is to be self-supporting, there is no other source of funds from which the final death claims can be paid.

In theory, of course, the aggregate reserve cannot be allocated to individual policyowners. Insurance is a group proposition, and the reserve is a group concept. To be sure, the fund is held to guarantee performance under individual contracts, but it is computed on a group basis. If an occasion should ever arise whereby the total reserve fund would have to be apportioned among the various policyowners, as in the event of liquidation, such apportionment should, strictly speaking, take into account the relative state of health of the various policyowners and their respective chances of survival. A policyowner in poor health would be entitled to a relatively greater share of the aggregate reserve than one who is in good health since the value of his or her contract, as measured by the relative chances of death, would be greater. Apportionment on the basis of the relative value of the policies would clearly be impossible. For all practical purposes therefore the reserve under a particular policy is considered to be its pro rata share of the aggregate reserve.

The retrospective method of reserve valuation may also be illustrated with reference to an individual policy. Such an illustration, however, presupposes a familiarity with the cost-of-insurance concept, which was discussed briefly in chapter 2. There it was pointed out that under level premium insurance, the company is never at risk for the full face amount of the policy due to the fact that a reserve is created under the contract with the payment of the first premium, and if the policy remains in force, it is available for the settlement of any death claim that may arise. In the event of a policyowner�s death, the company returns the reserve under the contract and adds enough to it from sums contributed by all policyowners in the deceased�s age and policy classification�including a contribution from the deceased�to make up the amount due under the contract. The sum that each policyowner must contribute as his or her pro rata share of death claims in any particular year is called the cost of insurance. It is the amount the policyowner must pay for protection.

The cost of insurance is determined by multiplying the net amount at risk (face amount of the policy less the reserve) by the tabular probability of death at the insured�s attained age. Thus if at the end of 20 years the reserve under an ordinary life policy issued at age 30 is $264.27, the cost of insurance for the 20th year is $735.73 x 0.00956, or $7.03. The net amount at risk is $735.73, and the probability of death at age 49, according to the 1980 CSO Male Table, is 9.56 per 1,000. Therefore that policy�s share of death claims during the 20th year of insurance is $7.03 per $1,000 of the policy�s face amount.

The cost of insurance for a $1,000 ordinary life policy issued at age 35 is shown in table 16-2 (at the end of this chapter) for each policy year to the end of the mortality table. As might be expected, the figure increases each year through the 57th policy year, or age 91, at which point it amounts to $38.52. It then declines in each of the following 5 policy years since the net amount at risk is decreasing at a more rapid rate than the death rate is increasing. The mortality rate increases more rapidly than the amount at risk decreases for policy years 63 and 64, and the cost of insurance increases again in those years.

It is not until the 29th policy year, or age 63, that the cost of insurance exceeds the net level premium of $11.60 for an ordinary life policy of $1,000 issued to a male aged 35. This means that through the 28th policy year, each annual premium makes a net addition to the policy reserve.

In applying the cost-of-insurance concept to the determination of a retrospective reserve, it is necessary to know the terminal reserve for the policy year in question�the very value that is being sought. This poses no problem if the computation is being performed algebraically, but it creates a mathematical impasse if the reserve is being computed arithmetically. To avoid the introduction of algebraic symbols, this arithmetic contradiction will be ignored in the following illustrations.

The net level premium for an ordinary life policy of $1,000 issued at age 35 is $11.60433. Invested at 4.5 percent interest, this sum will amount to $12.13 at the end of the first year. The policy�s share of death claims during the first year is $2.09 ($989.96 x 0.00211), which, deducted from the accumulated sum of $12.13, leaves $10.04 as the terminal reserve for the first year. This sum will be supplemented at the beginning of the 2d policy year by the second net level premium of $11.60, producing a sum of $21.64 that will be invested at an assumed rate of 4.5 percent interest throughout the year and will earn $0.97. Thus a fund of $22.61 will have accumulated by the end of the 2d year, from which $2.19 will be deducted as the cost of insurance ($979.58 x 0.00224), producing a 2d-year terminal reserve of $20.42. This process continues until, by the end of the 10th policy year, the reserve is $115.41. The accumulation during the 11th year then takes the following form:

Terminal reserve for 10th year $115.41

Add: net level annual premium 11.60

Initial reserve for 11th year $127.01

Interest earnings at 4.5 percent 5.72

Fund at end of 11th year $132.73

Deduct: cost of insurance 3.96

Terminal reserve for 11th year $128.77

It may be said that the retrospective terminal reserve for any particular policy year is obtained by adding the net level annual premium for the year in question to the terminal reserve of the preceding year, increasing the combined sum by one year�s interest at the assumed rate, and deducting the cost of insurance for the current year. If the policy is paid up or was purchased with a single premium, there will be no annual premiums to consider, and the cost of insurance must be met entirely from interest earnings on the reserve. The terminal reserve for the 11th year under a 10-payment life policy issued at age 35 in the amount of $1,000 would be obtained as follows:

Terminal reserve for 10th year $303.19

Interest earnings at 4.5 percent 13.64

Fund at end of 11th year $316.83

Deduct: cost of insurance 3.12

Terminal reserve for 11th year $313.71

If an ordinary life policy issued at age 35 becomes a death claim during the 11th policy year, it contributes a total of $132.73 toward the payment of its own claim, leaving only $867.27 to be contributed by other policyowners. If the policy is still in force at the end of the year, it contributes $3.96 toward the payment of death claims under other policies in its classification, leaving $128.77 as the terminal reserve. Note that the policy makes a contribution toward the cost of insurance whether it becomes a claim or not, a fact that has been alluded to earlier but not statistically demonstrated. It is interesting to note the difference in the 11th-year cost of insurance between the ordinary life ($3.96) and 10-payment life ($3.12) policies, which is attributable solely to the difference in the reserves or, conversely, the amount at risk.

Although the retrospective method of computation clearly demonstrates the origin and purpose of the reserve, its use is not preferred by actuaries. It is seldom used in actual reserve calculations for scheduled or fixed-premium policies. (As noted earlier, however, it is the method often used for flexible-premium policies.) State valuation laws invariably express reserve requirements in terms of the prospective method. This does not mean that a company must use the prospective approach since any method that will produce reserves equal to or in excess of those that would be derived by the statutory formula is acceptable. Nevertheless, insurers tend to prefer the prospective method because of its simplicity.

The prospective reserve, V, under a policy issued at age x, at the end of any given number of years, t, is equal to the net single premium for the policy in question at the age of valuation, A_x+t, minus the net level premium at age of issue, P_x, multiplied by the present value of a life annuity due of $1 for the balance of the premium-paying period calculated as of the age of valuation, represented by ä_x+t. The full net level reserve 10 years after issue for an ordinary life contract for $1,000 issued to a male at age 35 is therefore determined as follows:

At age 35 the net single premium for an ordinary life policy is $212.27, and the present value of future net level premiums at the inception of the contract is likewise $212.27. There is no reserve at this point. The net single premium for an ordinary life policy issued to a male at age 36 is $220.18, while the present value of future net level premiums for a policy issued at age 35 is only $210.14. Therefore a terminal reserve of $10.04 comes into existence at the end of the first policy year. By the end of the 10th policy year, the present value of future net level premiums will have declined to $187.78 ($11.60 x $16.18157), while the net single premium for an ordinary life policy at the beginning of the next year�that is, at the insured�s age 45�will have risen to $303.19. The difference of $115.41 must be on hand in the form of a reserve. At age 65 the net single premium for an ordinary life policy is $557.75, while the present value of future premiums at the age-35 rate is only $119.17. The reserve at the end of the 30th policy year therefore is $438.58. The two sets of values continue to diverge until at age 100, when the present value of future premiums is zero because there are no more premiums to be paid, whereas the net single premium for an ordinary life policy at age 100�if one could be purchased at such an advanced age�would be $1,000 since the policy would immediately become a claim. Thus the reserve at age 100 would be equal to the face amount of the policy, a principle noted earlier.

The values for line AB remain the same for the attained ages shown for any whole life policy, irrespective of the age of issue, but the values for line AC depend on a specific age of issue since the net level premium remains constant. Hence the slope of the curves differs with different ages of issue and with different types of policies; nevertheless, for any permanent form of insurance, the disparity between the two sets of values will ultimately equal the policy�s face amount.

This method can be used to determine the reserve under any type of life insurance or annuity contract and for any duration. For example, the reserve at the end of 15 years under a 20-payment life policy issued at age 30 is $303.19�($13.26 x $4.544), or $242.93. The sum of $303.19 represents the net single premium for a whole life policy issued at age 45, A_x+t, while $13.26 represents the net level premium for a 20-payment life policy issued at age 30. In this case, however, the level annual premiums take the form of a temporary life annuity�namely, the present value of a temporary life annuity due for 5 years, computed as of age 45. This value is $4.55.

Under limited-payment whole life or endowment policies, the present value of the temporary life annuity due becomes zero after all premiums have been paid, and the reserve becomes the net single premium for the policy in question at the insured�s attained age. Thus the reserve at the end of 30 years under a 20-payment life policy issued at age 35 is the net single premium for a whole life policy issued at 65. This means that at any particular attained age the reserves under all paid-up whole life policies are identical (if they are based on the same mortality table and interest rate assumption), regardless of age of issue. The same is true of all paid-up endowment policies with the same maturity date. This principle is illustrated in figure 16-2, which shows the reserves under different forms of whole life policies issued at age 35.

In measuring its liabilities under outstanding contracts, whether accomplished retrospectively or prospectively, an insurance company must assume certain rates of mortality among its policyowners and a certain rate of earnings on the assets underlying the reserves. The assumptions as to mortality are reflected in the choice of the mortality table used in the valuation, while the appraisal of investment potential is reflected in the rate of interest selected for the computations. The reserve values heretofore cited are based on the 1980 CSO Male Table and 4.5 percent interest. However, other assumptions have been and are being used in reserve computations. It is important therefore to note the impact of the choice of basic actuarial assumptions on reserves.

There is a widespread belief that the higher the rates of mortality assumed in a reserve computation, the greater the reserve will be. The fact is that the level of mortality, per se, does not determine the size of the reserves. A mortality table that shows a much higher death rate at every age than another table could produce a much lower reserve than that computed on the basis of the other table. The factor that governs the size of the reserve is the amount of bend of the mortality curve or the rapidity of increase in the rate of mortality from age to age. (Mathematically, the higher the second derivative of the mortality function, the higher the reserve required.) The sharper the bend in the slope of the mortality curve, the greater the reserve will be. This is true even though the steeper curve shows a lower rate of mortality at all ages up to the terminal age. This principle is illustrated in a crude fashion in figure 16-3.

Curves AB and CB represent the death rates from ages 25 to 100 under two hypothetical mortality tables, one reflecting unrealistically high rates of mortality and the other showing more normal rates. Lines DE and FG represent level premiums and are placed at such distances above points A and C as to create triangles beneath the broken lines, in each case approximately equivalent in area to the triangles that can be formed above the dotted lines. This obviously ignores the influence of compound interest and grossly exaggerates the redundancy of the level premiums in the early years of the insured�s life, but it graphically illustrates the effect of leveling out mortality curves of differing slopes. It is apparent that the mortality table represented by curve AB would produce the higher reserves.

In practice it is not always possible to predict visually the relative magnitude of reserves under different mortality tables at various ages and durations. A particular mortality table may show higher death rates at some ages and lower death rates at other ages, compared to another table. In such event the reserves under the specified table may be higher at some ages and durations than those of another table and lower at other ages and durations. For example, a select mortality table, which shows a rapidly increasing death rate during the early years of coverage as the effect of selection wears off, produces larger reserves for short durations than a table that does not exhibit such a sharp increase in death rates.

Unlike a change in mortality assumptions, which may produce either an increase or a decrease in reserves depending on age of issue and duration, a change in the interest assumption will affect the magnitude of reserves in the same direction�but not necessarily to the same extent�at all ages of issue and durations. Specifically, a decrease in the assumed rate of interest increases reserves, while an increase in the rate decreases reserves.

The impact of a change in interest assumption on reserves is not easily explained in terms of the conventional retrospective and prospective approaches. For example, it might be concluded, retrospectively, that a reduction in the assumed rate of interest would result in the accumulation of a smaller reserve at the end of any given period short of the maturity of the contract or the end of the mortality table. Such a conclusion, however, ignores the fact that the lower interest assumption implies a larger net premium, and the problem becomes one of measuring the effect of accumulating a series of larger net premiums at a lower rate of interest. Inasmuch as the augmentation of net premiums more than compensates for the loss of interest earnings involved in the lower interest rate, a larger reserve results. On the other hand, an increase in the assumed rate of interest would require a smaller net premium, offsetting for a time the influence of the higher rate of interest earnings and producing a reserve that is smaller at all durations up to the end of the mortality table or earlier maturity of the contract than that computed on the basis of a lower interest assumption.

Prospectively, the analysis is complicated by the fact that the present value of future benefits and the present value of future premiums are affected in the same direction by a change in the interest assumption. However, inasmuch as the present value of future benefits is always influenced to a greater degree than is the present value of future premiums, a decrease in the assumed rate of interest always produces a larger reserve, and an increase in interest produces a smaller reserve.

The simplest explanation revolves around the fact that the reserve must accumulate to the face amount of the policy by the end of the mortality table for a whole life contract or by the maturity date for an endowment policy. Therefore the lower the rate of interest at which the reserve is to be accumulated, the larger the fund to which the lower rate is to be applied must be at any given time. To take a very elementary example, if $1,000 must be accumulated by the end of any particular year and only 2.5 percent interest can be earned on invested funds, a sum of $976 must be on hand at the beginning of the year. If a yield of 3 percent can be realized, only $971 need be on hand at the beginning of the year.

During the decline in interest rates during the 1930s and 1940s, many life insurance companies reduced the interest assumption in computing reserves under old contracts, which, of course, meant that the reserves under such contracts were increased. This was accomplished by transferring funds from the surplus account to the reserve account (at one time or over a period of years) or by diverting to the reserve account a portion of current mortality, interest, and loading savings that would otherwise be distributed as dividends. Once the necessary funds were transferred to the reserve account, the policies thereafter were treated as if they had been written on the lower interest basis in the first place, except that surrender and loan values were usually not increased. Insurers were then in a position to meet their reserve requirements with lower earnings on their invested funds.

In some states, notably New York, a reserve-strengthening program of this sort can be undertaken only with the insurance commissioner�s consent, and once it has been carried out, it can be reversed only with official consent. This requirement is in the interest of preserving equity among the various classes of policyowners.

It is clearly possible that similar reserve strengthening may again be required in the future if the economy ever sustains a long enough period of low investment returns. The investment returns available in the early 1990s were often alarmingly close to the guaranteed rates in the contracts. If investment returns remain at or below the level of contract guarantee rates long enough for the insurer�s entire portfolio to be reinvested, there will be no investment returns to supplement the low rates on new portfolio investments. Such a scenario could force life insurers to strengthen their policy reserves.