The chief difference between deriving gross participating premiums and gross nonparticipating premiums is the set of actuarial assumptions used. Conservative assumptions produce higher margins. In participating premiums these margins can be returned as dividends when they emerge. In nonparticipating premiums the lack of dividends may make high premiums particularly noncompetitive. Asset share calculations test for this possibility.

Two basic approaches are available to derive a gross nonparticipating premium. Under the first the actuary computes the premium based on the most probable assumptions about mortality, interest, expenses, and terminations, plus a specific addition for profit. The principal advantage of this approach is the explicit provision for profits for each plan of insurance and each age of issue.

The second approach uses assumptions that are more conservative than the most probable, but it makes no specific allowance for profit. Profits must emerge from more favorable experience than was assumed. Profit that varies widely among the various plans and issue ages is regarded as an advantage by some on the grounds that profit depends on the risk assumed. This second method is sometimes used alongside the first to show how large a profit margin is needed to assure the company a minimum return if deviations from the most probable assumptions occur.

The premium computation shown in the following pages uses the first approach.

Of the four assumptions affecting asset shares�mortality, interest, expenses, and terminations�only the impact of interest increases with duration. The impact of interest increases with the size of reserves and thus has its biggest impact at later durations, where it is most difficult to predict.

The impact of mortality and terminations decreases over time because the amount at risk decreases as the duration increases. The impact of expenses decreases because they occur primarily in the early years. While current mortality rates are easy to find, favorable and unfavorable changes in the future due to medical advances, underwriting changes, and social changes are difficult to project. To ignore the impact of medical advances is conservative for insurance rates but careless for annuity rates.

Economic changes greatly affect policy termination. For example, changes in unemployment and alternative investment opportunities can influence termination. Termination rates also vary from company to company, from plan to plan, and with issue age.

We illustrate the process of setting gross nonparticipating premiums by calculating and testing the gross premium for a 10-payment ordinary life policy issued to a woman aged 32. This illustration uses an asset share calculation to test a tentative premium. The principles involved apply equally to the testing of a tentative participating premium.

The tentative premium can be quite arbitrary. When calculations are part of a general rate revision, the tentative premium is usually the current premium. Alternatively, it may be the premium charged by a competing company. For our illustration, the tentative premium is the 10-pay net level premium based on 1980 CSO Female Table mortality and 5.5 percent interest. For $100,000 of insurance, this is $1,451.57.

Our illustration (see table 17-2) uses expense factors consistent with those in table 17-1.

One additional set of values is needed before we begin the asset share calculation�the surrender values available each year to terminating policyowners. These values for the first 10 policy years under a $100,000 10-payment life policy issued at age 32 are assumed to be as shown in table
17-6.

Assumptions about the following provide the raw materials to calculate the asset share:

These assumptions combine as shown in table 17-7 to produce the asset share per $100,000 at each duration. (See pages 462�463.)

According to table 17-7, 10,000 policyowners will pay a total of $2,777,222 in effective premiums at the beginning of the first year. These funds earn interest throughout the year at 5.5 percent and grow to $2,929,969 at the end of the year. Death claims will diminish this amount by $391,091�$380,000 in claim payments, $760 in settlement expenses, and $10,331 in loss of assumed interest. The loss of interest arises because we have assumed that the total effective premiums will accrue interest to the end of the policy year. On average, however, claims will be paid at mid-year, so the company loses interest earnings on these payments. It is also assumed that 1,999 policyowners surrender their policies at the end of the first year, receiving $0 individually and in total.

At the end of the year a fund of $2,538,878 is assumed to be on hand. Dividing this pro rata among the 7,997 surviving and persisting policyowners gives each $317. That is the asset share per $100,000 at the end of the first year. The terminal reserve at each duration is shown for comparison. Observe that at the end of the first year the reserve exceeds the asset share by $1,071 and the asset share exceeds the surrender value by $317.

At the beginning of the second year, 7,997 surviving and persisting policyowners pay a total of $10,783,958 in premiums. When added to the fund from the end of the first year, this produces a total fund at the beginning of the second year of $13,322,837. At 5.5 percent interest, this fund amounts to $14,055,593 at the end of the second year before deduction of death and surrender claims. Death claims and settlement expenses, adjusted for loss of interest, and surrender payments reduce the fund to a net balance of $13,307,374. Divided pro rata among the 7,194 surviving and persisting policyowners, this fund yields an asset share of $1,850. This falls short of the second-year terminal reserve by $1,001 per $100,000, and surpasses the surrender value by $1,367.

This process continues through the next 8 years. By the end of the 10th policy year, the fund is seen to have grown to $77,302,999, an amount sufficient to provide $16,032 to each of the 4,822 surviving and persisting policyowners. This is $1,512 less than the full net level premium reserve and the surrender value at that point.

By comparing the asset share at each duration with the comparable surrender value and reserve, the company can evaluate the appropriateness of the trial gross premium. Until the asset share equals or exceeds the surrender value, each termination is a direct drain on the company�s surplus. In other words, the company gives back to each withdrawing policyowner more money than that policyowner has contributed to the company. This situation may prevail for several years under many plans and ages at issue. The asset share is usually negative during the first few years of a continuous premium whole life policy, it takes several years to exceed the cash value, and takes even more years to exceed the reserve. However, under a high-premium policy, such as the 10-payment life shown in table 17-7, the asset share normally should exceed the surrender value by at least a small amount even at the end of the first year.

A more fundamental test companies use is the period required for the asset share to equal or exceed the full net level premium reserve. Until that occurs, the company has not recovered its acquisition expenses and is still showing a book loss for the block of business represented in the asset share calculation. Once the

asset share exceeds the reserve, acquisition expenses have been recovered in full and that block of policies is contributing to the company�s surplus. Based on many considerations a company decides how long it can afford�and is willing�to wait before recovering its outlay. That period of time is called the policy�s validation period. If the company can wait 10 years to recoup its acquisition expenses, it uses gross premiums that accumulate asset shares exactly equal to the reserves at the end of the 10th policy year.

It would be sheer accident if a trial gross premium produced an asset share precisely equal to the full net level premium reserve at the end of the validation period. Variation will exist in one direction or the other. To eliminate the variation, the trial gross premium is adjusted either upward or downward. The amount of this adjustment is found by dividing the difference between the asset share and the reserve at the end of the validation period by the future value of a $1 premium increase, and adding (or subtracting) the result to the trial gross premium.

The technique is illustrated, continuing with the example of the 10-payment life policy, by assuming a validation period of 10 years. A 10-pay life policy will have settlement costs and maintenance expenses after premiums have ceased. The present value of these costs, $350, is added to the 10th year reserve when determining a gross premium. The fact that the asset share is $1,512 less than the reserve at the end of the validation period means that the trial gross premium is high enough to make the policy profitable by the end of the second year. To find the required correction, the effect of changing the trial gross premium by $1 upon the accumulation at the end of 10 years is measured. Then, by simple proportion, the exact change in the premium that would increase (or decrease) the accumulation by the desired amount can be found.

It is helpful at this stage to visualize the change in the trial gross premium as an increase of $1, whatever the direction of the adjustment needed. (See table 17-8.) Then it will be apparent that this additional annual $1 payment will not have to bear any share of the death and surrender claims, since these were met through the original premium payments. However, the additional $1 should bear its proportionate share of expenses that vary directly with the size of the premium. Since these expenses were earlier assumed to be 54 percent of the first-year premium and 4 percent of the renewal premium, the effective additional premium will be $0.46 the first year and $0.96 for each of the other 9 years. The number of surviving and persisting policyowners at each duration remains the same, and so does the assumed rate of interest earnings. Therefore the additional premium of $1 will bring in an additional sum of $4,600 the first year. At 5.5 percent interest, that additional amount accumulates to $4,853 by the end of the first year.

Because of lower renewal expenses, the additional effective premiums for the second year will aggregate $7,677, which, when supplemented by the fund at the end of the first year and interest at 5.5 percent on the composite fund, amounts to $13,219 at the end of the second year. By the end of the 10th year, the additional premium of $1 paid each year by the surviving and persisting policyowners will accumulate to $78,667. Divided pro rata among the 4,822 policyowners surviving and persisting at that point, this sum would provide an additional $16.32 increment to the asset share for each policy. In other words, increasing the trial gross premium by $1 increases the asset share at the end of 10 years by $16.32.

Table 17-8 on the following page presents the calculations supporting this adjustment.

When constructing a set of gross premium rates, the detailed study just outlined would not be done for each policy and issue age. This would be too much work and would probably produce inconsistent rates. Rather, the procedure would be used for rates at quinquennial or decennial ages. These rates would be tested for adequacy for a simplified portfolio using policies with only these ages. The simplified book of business is selected to represent the company�s overall book of business and is called a model office.

The asset share calculation is concerned only with the adequacy of the proposed gross premiums. Once this has been established, the gross premiums at the representative ages are compared with those of other companies operating in the same territory to learn whether the rates are competitive. A company whose rates are clearly out of line will have trouble holding its agents and obtaining

new business. If the survey of other companies� rates shows the test set to be competitive, the company proceeds to derive the complete set of rates by formula or interpolation. If rates appear to be too high relative to those of the competition, the company considers adjusting its rates.

If the "most probable" mortality, interest, expense, and termination rates have been used in calculating gross premiums, there is little possibility that the competitive situation can be improved by changing any of those assumptions. Even with the same basic assumptions, however, the premiums can be reduced by extending the period over which acquisition expenses are amortized. This increases the drain on surplus, but it may be the most practical solution. If a specific allowance for profit has been made in the calculations, shaving this margin may reduce premiums slightly. If extending the validation period and narrowing the profit margin do not produce competitive premiums, more fundamental adjustments may be needed. Such adjustments might include more stringent underwriting requirements, less conservative (and thus higher yielding) investments, greater operating economies, or elimination of less persistent policies.

The final step in deriving a set of gross premium rates is to review the results for consistency among the various plans and ages at issue. Identical premiums should always be charged for identical benefits. The premium scale should contain no "bargain rates" since such rates are likely to attract an undue volume of business for certain plans and ages of issue, which may indicate the presence of a high level of adverse selection.

As mentioned earlier, the technique underlying the calculation of nonparticipating gross premiums in larger stock companies is used by some mutual companies. It is modified to reflect the payment of policyowner dividends. In using this technique, a mutual company

At this stage the computation does not allow for dividend distributions. The margins in the basic assumptions are so narrow that no funds are presumed to be available for distribution to policyowners. Trial gross premiums, adjusted to reflect the redundancy or deficiency in the asset share, are compared with the gross premiums (after dividends, in the case of participating gross premiums) of competing companies. Once the premiums have been fitted to the best competitive advantage, the company considers its dividend policy.

A dividend scale of any desired level and pattern can be developed without relating the resulting dividends to any particular sources of surplus. Margins to support the proposed dividend scale are added directly to the gross premiums. Usually several sets of premium rates, based on various assumed margins, are constructed and compared before the final set of representative gross premiums�with the built-in dividend scale�is selected. Gross premiums for all ages of issue usually are computed by loading the valuation net premiums in accordance with a formula that experimentation has found to develop rates approximately equivalent to the desired gross premiums when applied to net premiums. As pointed out earlier, such a formula usually is the sum of a percentage of the gross or net premium, a constant amount per $1,000 of insurance, and a policy fee.

Some companies vary the procedure by calculating the trial gross premiums on a nonparticipating basis and then adding the margin for a predetermined dividend scale before running the various asset-share tests. Dividend distributions operate as a decrement in such a procedure, along with death claims and surrender payments. Each approach produces the same results.

So far the calculations have assumed premiums to be paid annually. Theoretically, premiums paid more frequently than once per year could be calculated in precisely the same manner by using probability and interest functions based on shorter time units. To derive true monthly premiums, a mortality table that shows the rate of mortality month by month, rather than annually, is needed. Similarly, claim payments could be discounted monthly instead of yearly. Such precise computations are not used in practice, however, since mortality studies produce annual rates. Instead, actuaries calculate monthly, quarterly, and semiannual net premiums in a way that distributes deaths uniformly between whole ages. That technical procedure, which must include loadings for the expense of additional premium processing, is not covered here.