to the
Theory of
Life Insurance Reserves
by
Dr. August Zillmer
Actuary, Germania Stock Company
Stettin, Prussia
Stettin, 1863
Press of Theodore von der Nahmer
Translated by
Bill Roach and Gunnar Alksnis
Spring, 1989 Copyright
Translators' Note
All of the financial figures of the original paper have been reproduced in the translation. This has been achieved by substituting dollars for the Prussian taler. The units of Prussian currency at the time were as follows:
30 Silbergrosschen per Taler
12 Pfennig per Silbergrosschen
A single taler in 1861 was worth $.75 in U.S. currency. Reconstructions
of the Consumer Price Index for that era show a CPI of 27 for 1861 where
the CPI of 1967 is 100. All of the tables in the origina paper have been
recalculated using the spreadsheet software Quattro. Except where indicated,
the only differences between the recalculated values and the original values
are due to rounding.
Introduction
More than any other corporations life insurance companies should be held to exact and strict accounting rules. The entry into a life insurance contract is an act of self denial and an example of the noblest love of spouse or child. Often the policy is the only inheritance and the total estate of widows and orphans. Since life insurance institutions draw in a considerable portion of the savings of thousands of policy holders on the basis of contracts several of which will not expire before a half or sometimes almost a whole century. Therefore, their solidity and solvency must be mathematically provable and indisputable.
These undeniably true words which were spoken by the New York State
Insurance Commissioner when he withdrew the license of the American Mutual
Life Insurance Company of New Haven, Connecticut, to do business in the
State of New York. In honor of these words, in the following pages we will
present several considerations about two different methods of calculating
the reserves of life insurance institutions.
In the year 1857, Mr. Wilhelm Lazarus , in a thorough article (See Rundschau
der Versicherungen, 1857, p317.) had already called attention to the
misleading principle of estimating the present value of obligations at
a lower interest rate than the present value of the yet to be collected
premiums and demonstrated the consequences of such misleading calculations
on the balance sheet of a company which thus calculates. Yet since that
time, practically nothing has happened to prevent the use of this misleading
principle. Here and there, one finds in professional journals the remark
that the reserve figured on gross premiums is too small, but that is all
one finds. In more recent times, prompted by the publication of the balance
sheets of some English companies, some voices have been raised against
the mischief of these calculations. With this, other aspects of the life
insurance business have been drawn into the conversation and, here and
there, mistaken concepts have become widely shared. For example, it was
asserted that the recent custom of paying agents a high commission for
the acquisition of a life insurance contract, about 1 % of the insured
amount, seems to make it impossible to set up an actuarially sound reserve.
And thus companies that pay such commissions are forced to calculate reserves
too low, if not even according to the principle of gross premiums. The
presentation of such mistaken, when not intentionally misleading assertions,
are well suited to deeply affect the trust of the public in life insurance
companies and will harm exactly those companies that have published their
business details with the greatest openness and make no secret of their
commissions. We attempt to contribute, to the best of our ability, the
correct view about the reserves of life insurance companies and in the
first segment of the following monograph provide the proof that even for
the custom of a closing commission, a reserve can be established that has
a completely calculable magnitude. In the second segment, we will briefly
discuss the method of calculation with gross premiums, and we believe that
we thereby show that many phenomena existing in the real world are natural
consequences of this misleading system.
I.
In a life insurance institution, the reserve plays a large role. The
annual premium for an insurance policy [in this we have in mind the usual
whole life insurance policy with annual premium payment] remains constant
for the whole term of the insurance policy while the risk that the insured
amount becomes due by the death of the insured grows with each year. And
if an insurance policy were always issued for one year only, it would require
an increase in premium each year. Therefore a premium that remains constant
for the whole term of the insurance policy is greater for the first year
after the issue of the insurance policy than the yearly renewable term
(YRT) premium; later the premium is smaller than the YRT premium. If a
certain number of persons enter an insurance bank, then the bank will create
a considerable surplus after covering the occurring cases of death among
those persons. And in the later years this surplus has to cover the added
expenditures when the bank must pay more for cases of death than it receives
in premiums. For example, the attached Table II shows that a bank which
insures a number of 30 year old persons with a sum of $500,000, collects
more in premium during the first 25 years than it has to pay out for cases
of death. From the 26th year on, the expenses for deaths overtake the premium
income. For example, in the 40th year, the excess of the death claims over
the premiums is $8,679.
Therefore as the compelling consequence of the level premium method,
a life insurance bank must set up a reserve fund. In each case, the amount
of the reserve fund cannot be arbitrarily set as it suits the bank, but
it has to be mathematically and precisely determined, just as the annual
premium is calculated mathematically and precisely on the basis of a mortality
table and a certain rate of interest. For example, the insurance bank may
not lower the reserve fund, speculating that the mortality among the insured
will be more advantageous than could be expected from the mortality table
used in the calculation of the premium or that it will, in consequence
of fortunate speculations, obtain a higher interest income with their monies
and thus be easily able to cover the shortage in the reserve funds. The
reserve fund must be set up much more from the paid in premiums of the
existing insurance policies and the calculable interest after deduction
of the same calculable future expenses for death, and to be sure the calculable
expenses, not the actually caused ones for otherwise a sustained excess
mortality among the insureds could, if not exhaust the reserve fund, at
least significantly lower it below the level required for the future performance
of the bank. The reserve fund is not to be set up from the premiums paid
by the insured but from the net premiums. The premiums arrived at by exact
calculation (net premiums) are to be increased by a loading which should
reimburse the bank for business expenses and the risk that mortality among
the insureds exceeds the expected limits. The loading, the size of which
will be established in the individual banks according to more or less arbitrary
rules of thumb, goes to the bank, and the reserve fund is calculated from
the remaining net premiums after the deduction of this loading. As soon
as the net premium described above is determined, the reserve will result
as a mathematical consequence at every point of time during the term of
the insurance policy.
As a rule the net premium is so established that it remains constant
for the whole term of the insurance policy, just as the loaded premium
paid by the insured remains constant. But clearly there is no reason not
to include other considerations relating to the amount of net premium if
only these are rationally calculated, completely covering the obligations
of the bank. There are insurance policies with rising or falling premium
payments whereby naturally not only the loaded premium changes in the different
periods during the term of the insurance policy, but also the net premium.
To arrive at that firmly established net premium that we believe we must prefer in general and prefer especially in the current situation in the business, we put forward several considerations:
The participation of the German public in life insurance has increased in a gratifying fashion in recent years. Not only do some of the few older German companies do more significant business than in earlier decades, but also the newer German companies and besides them a considerable number of foreign companies work with generally growing success. The Germania has even experienced such a spectacular participation by the public that last year (1862) it issued insurance contracts in a sum of significantly more than $6 million and in the month April of this year alone, in a sum of more than $1 million. This glorious upswing in the life insurance business is not only the consequence of the correct appreciation and high significance which life insurance has from the individual, the family, and the state and which is permeating all levels of society, but this upswing is also brought about by the efforts of the executives of the life insurance companies and their agents. These executives introduced the practice of paying the agent a higher commission at the issue of the insurance policy [previously the agent had only received a commission which remained constant for the term of the insurance policy. The agent kept a portion of the premiums he collected and thus achieved for himself a worthwhile income only after years and years]. This succeeded in attracting active business men as agents who then, being placed in a position to quit their other business, were able to use their whole strength for the attraction of new insurance policies. The success, that the introduction of this commission practice has had until now is its own best justification. But complaints against the commission method force another question: how does a company defray the agency costs which must grow proportionately for a large scale addition of new insurance policies. We admit that if a company is required not only to post an actuarially sound reserve [naturally this must be required] but if it is also required to set the amount of the reserve on the basis of the determined, prescribed, or traditional calculation model (net level premium ), then company can thereby run into difficulties. The expenses of an insurance policy, for the coverable mortality for the first year and for the reserve set aside at the end of the first year according to any one calculation model can and must under all circumstances be greater than the premium income for this insurance policy. The thus growing deficit will naturally be the greater, the greater the increase in new insurance policies is, and when the increase is sufficiently large, the deficit too will become so large that even a company with significant means could run into difficulties. It should be clear even to the lay person that a contradiction emerges here. On the one hand, one wishes ample increases in new insurance policies; on the other hand, one would be financially embarrassed by this and this could happen even [as will be demonstrated in the following] with commission rates that have not been set too high.
It would be wonderful if a solution for this could not only be found but if it offered itself, and this is really the case. The obvious thought is that under the old, usual method of calculation which at the close of an insurance policy posts the costs growing out of that closing, possibly in a prepaid commission account, and this account is entered as an asset of the bank. The sums brought into this account in an individual year would then have to be amortized in a certain number of years, or what would be preferable is that from the total amount of the prepaid commission each year a certain and adequate portion of the premium income would be written off. This model was publicly discussed on April 26 of this year in an article in the German insurance paper of Breslau, but for various reasons we cannot sanction this. For one, there is no measure and no limit for the amount of those costs that could be placed in this prepaid commission account. In essence this account would have the effect of lessening the base capital or paid in the reserve; and then it would certainly be possible to endanger the stability of the company through excessive expenses for the acquisition of new insurance policies. Second, the artificial asset of prepaid commissions should really be gradually amortized. Since this requires a relatively high fraction of the annual premium, this means that the coverage of the other costs and the gain for a long series of years, respectively, are unnecessarily lowered. Finally, thirdly, there could and would be found, in the account of the prepaid commissions, sums prepaid for such insurance policies that have lapsed. Later insureds then have to pay subsequently the costs which have been incurred by earlier insureds.
In the following simple way one develops a model which is completely
untouched by these disadvantages. One calculates the net premium for an
insurance policy in such a way that in the first year it is smaller by
a certain amount than for the following years. We omit the development
of the necessary formula here. It is, by the way, extremely simple. We
get our net premium as the usual premium which remains constant for the
whole term of the insurance policy. Take the amount by which the first
annual premium should be smaller and divide it by an annuity due for the
issue age and add it to the usual net premium.
For example, calculating with 3 1/2 percent interest and using the Combined
Experience Table, the net annual premium at issue age 40 for an endowment
for $100 at age 90 [and in the following, this is what we always have in
mind] is 2.484 percent. Should the first annual premium be smaller by 1
percent of the insured sum, then we get as the premium for the following
years 2.484 + 1 divided by the annuity due at age 40 which ends at 90,
that is, 2.484 + 0.059 and this as premium for the first year 2.543 - 1.
= 1.543. In contrast, if the first premium should be smaller by 1 1/4 percent
of the insured sum, then the first premium is 2.484 + 5/4 * 0.059 - 1.25
= 1.308 and the premium for all the following years is 2.484 + 5/4 * 0.059
= 2.558.
Naturally also the insured pays the same premium in the first year which
he will have to remit in the following years, and the bank covers the cost
arising from acquisition of the insurance policy with the portion of the
premium available to it in the first year. The reserve will be calculated
just as with the usual premium, that is, one finds the reserve for an insurance
policy which has been in existence for a number of years by deducting from
the net single premium at the attained age the product of
the net annual premium and an annuity due at the higher age. The reserve will be somewhat smaller here than according to the previous method, but exactly fulfills the demands:
If we solve the resulting equation, we find X, that is, the maximum of the costs that may be incurred at the closing of an insurance policy equal to the difference between the usual net premium and the premium for one year of term insurance, this difference multiplied by the annuity due at the issue age and divided by the annuity due lessened by one or immediate annuity at the same issue age.
The formula for the maximum of the closing costs can be expressed in
many ways. So this maximum, for example, is equal to the quotient of the
annuity due at the issue age lessened by one and the annuity of the next
higher age lessened by one, that is, for the insurance sum one, or it is
equal to the difference of the usual net premium at a one year higher age
and the usual net premium of the issue age multiplied by the annuity due
at the issue age or, and this is the simplest expression, the maximum of
the closing costs is equal to the net premium for the one year higher age
lessened by the premium of the issue age for a one year term insurance
policy. This simple form for the maximum of the closing costs can also
be directly deduced. The maximum that the company can spend for the acquisition
of an insurance policy is evidently reached when at the end of the first
year, the reserve is zero. The latter is the case if we take as net premium
the usual net premium of the next higher age. Since the premium for the
one year term insurance policy is used to cover mortality, there remains
for the coverage of costs caused by closing at most the difference between
the usual net premium of the next higher age and the premium for the one
year term insurance policy.
The following table shows the maximal rates for the closing costs if
one calculates according to the Combined Experience Table and 3 1/2 percent
interest.
| Issue | Maximum Rate of Closing Cost | Maximum Rate of Closing Cost |
| Age x | Percent of Insured Sum | Percent of Premium |
| 10 | 0.48 | 42.4% |
| 15 | 0.59 | 46.7% |
| 20 | 0.71 | 50.1% |
| 25 | 0.86 | 53.3% |
| 30 | 1.04 | 56.0% |
| 35 | 1.27 | 58.5% |
| 40 | 1.58 | 61.1% |
| 45 | 1.95 | 62.3% |
| 50 | 2.33 | 60.2% |
| 55 | 2.77 | 57.0% |
| 60 | 3.28 | 52.8% |
| 65 | 3.80 | 47.2% |
It follows from this table that if the average issue age is 37 years
or a little more, closing costs of a little more than 1 3/8 percent of
the insured sum can be applied; if we take into consideration that the
maximum rate for higher ages grows in an increasing relationship and that
furthermore the younger issue ages purchase smaller sums of insurance,
then we may assert that an insurance company which figures according to
the Combined Experience Table and an interest rate of 3 1/2 percent can
establish 1 1/2 percent of the insured sum as the average maximum that
it may spend for the acquisition of an insurance policy and that it may
cover by a corresponding calculable portion of the net premium.
It does not follow from this that 1 1/2 percent of the insured sum has
to be figured as the closing cost; on the contrary, I hold it advisable
to stay with 1 percent or 1 1/4 percent of the insured sum. It is possible
for the average issue age to decrease if younger persons participate in
insurance in greater numbers or if younger persons insure themselves with
larger amounts than the present experience has shown. Then it is possible
for the results of the calculation to develop so that the average issue
age decreases and the calculable rate of closing costs has become too high.
Since even in the case where we calculate the closing costs with only 1 - 1 1/4 percent of the insured sum, with high probability but not with absolute certainty we are secured against the danger that could emerge from a decline in the average issue age; therefore I consider a precautionary measure necessary which will be discussed in the following:
If we figure as closing costs more than the maximum rate for the youngest issue age, then at the younger ages the net premium to be used for the first year for the risk is smaller than the premium for a one year term insurance policy at the issue age or, in other words, the reserve at the end of the first insurance year becomes negative and because of this the net premium to be used in following years would be greater than the usual net premium for an issue age one year higher. If the company includes the negative reserve for a younger age in the total sum of the reserves this lowers the total reserves exactly by as much as the company has too little for the calculable mortality in the course of the year [of course, drawing interest until the end of the year]. The higher the closing costs are calculated, the greater will be the amount of the negative reserves and by so much more would the total sum of the reserves be forced down thereby. Add to this the fact that the cost must be written off against the company's capital for those insurance policies which lapse after a 1 year existence. In consideration of these circumstances, I hold it essential that a company does not include the negative reserves in the total reserves, but calculates the reserve for the end of the year as zero. From this grows for the company an expense which exceeds the calculable expense allowance. But this is easy to handle, as will be demonstrated, so long as we can assume that the calculable closing costs have not been set too high. If the conditions change so that it appears that the closing costs calculated up to now have been set too high, this will become evident when the portion of income that would be used for the formation of the profit is reduced while the premium reserve remains untouched and the company has, at the same time, a built in regulator for the magnitude of the closing costs.
The insurance polices that had a negative reserve at the end of the
first year are increasing more in the second year because of the consideration
of the calculable closing costs. If at the end of the second year, the
reserve should still be negative then instead of the negative value again
we set the reserve to zero and pay the calculable reserve only at the end
of the third year. This last happened only with the very lowest issue ages
as the following table shows:
| Issue | Net Level Premium Reserves | ||
| Age x | for a Policy for $100 | ||
|
Durations
|
1 | 2 | 3 |
| 15 | 0.584 | 1.187 | 1.809 |
| 16 | 0.606 | 1.232 | 1.878 |
| 17 | 0.630 | 1.279 | 1.949 |
| Zillmer Reserves @ 1 1/4 % Closing Costs | |||
| for a Policy for $100 | |||
|
Durations
|
1 | 2 | 3 |
| 15 | -0.659 | -0.049 | 0.581 |
| 16 | -0.636 | -0.002 | 0.651 |
| 17 | 0.612 | 0.045 | 0.723 |
In general, the expense that the bank incurs by not setting up the actual negative reserves but setting them to zero, that is, for some of the insurance policies the reserves are calculated as greater than they actually are, is not significant. If we calculate on the basis of 1 1/4 percent closing cost, the reserves will already be positive at the end of the first year for an issue age of 35 years; if we calculate on the basis of 1 percent, the reserves will already be positive for the issue age of 30. In general, the younger ages are covered with less insurance the lower the age so that where the closing expense for the individual insurance policy becomes somewhat larger, then there are only individual insurance policies that cause this expense. Add to this that in a steadily growing business, this expense is to be paid only in the first year. The net premium for the second year has to cover the mortality of that year and the negative reserve for the end of the first year and beyond that provide the positive reserve for the end of the second year. Since the negative reserve has already been defrayed by the bank, this portion of the net premium flows towards its profit or rather this portion defrays the negative reserves for the new insurance policies. In a business that grows at an increasing rate, the expense would repeat itself every year, and the increase would be the difference by which the insurance policies for the younger ages have grown more than in the previous year. To have grasp of the size of these costs we will make the specific and certainly not too low assumption that in a year bank insures
$10,000 for 20 year old personsThen if we calculate closing costs as 1 percent of the insured amounts, we would have
$20,000 for 21 year old persons
$30,000 for 22 year old persons
...
$100,000 for 29 year old persons
$120,000 for 30 year old persons
$140,000 for 31 year old persons
...
$200,000 for 34 year old persons.
|
|
|||
| Issue Age | Insured Sum | as Percent Insured Sum | in Total |
|
20
|
$10,000
|
-0.289
|
-28.9
|
|
21
|
$20,000
|
-0.263
|
-52.5
|
|
22
|
$30,000
|
-0.234
|
-70.3
|
|
23
|
$40,000
|
-0.205
|
-82.2
|
|
24
|
$50,000
|
-0.175
|
-87.5
|
|
25
|
$60,000
|
-0.143
|
-85.8
|
|
26
|
$70,000
|
-0.110
|
-77.1
|
|
27
|
$80,000
|
-0.075
|
-60.4
|
|
28
|
$90,000
|
-0.040
|
-35.9
|
|
29
|
$100,000
|
-0.002
|
-2.2
|
|
Sum
|
$550,000
|
|
-582.8
|
If we calculate closing costs as 1 1/4 percent of the insured amount
|
|
|||
|
|
|||
| Issue Age | Insured Sum | As Percent Insured Amount | In Total |
|
20
|
$10,000
|
-0.538
|
-53.8
|
|
21
|
$20,000
|
-0.511
|
-102.1
|
|
22
|
$30,000
|
-0.482
|
-144.7
|
|
23
|
$40,000
|
-0.453
|
-181.4
|
|
24
|
$50,000
|
-0.423
|
-211.5
|
|
25
|
$50,000
|
-0.391
|
-234.5
|
|
26
|
$60,000
|
-0.358
|
-250.6
|
|
27
|
$70,000
|
-0.323
|
-258.6
|
|
28
|
$80,000
|
-0.288
|
-258.8
|
|
29
|
$90,000
|
-0.250
|
-249.7
|
|
30
|
$100,000
|
-0.211
|
-253.0
|
|
31
|
$120,000
|
-0.170
|
-237.4
|
|
32
|
$140,000
|
-0.127
|
-203.1
|
|
33
|
$160,000
|
-0.082
|
-147.0
|
|
34
|
$180,000
|
-0.033
|
-66.8
|
|
Sum
|
$1,350,000
|
|
-2853.0
|
If a company in every year issues $550,000 more insurance inforce than
in the preceding year for the issue ages under 30, then it would have,
at 1 percent closing costs, an expenditure that exceeds the calculable
only by $586, that is, this expenditure does not even amount to 1/4 percent
of the premium income as soon as the premium income has climbed over $250,000,
or if the company figures on 1 1/4 percent closing costs, it would have
an annual expenditure of $2,870 when in every year it issues policies for
$1,350,000 more than in the previous year for the issue ages under 35,
thus less than 1 percent of the premium income when this has exceeded $300,000.
One could say that the company can figure the maximum rates for closing
costs for every age, the total reserve of the new insurance policy would
then amount to exactly zero, and the net premium of each individual issue
age would then become equal to the usual net premium of the next higher
age. If a company calculates its premiums from the start so that it every
time adds a certain percentile loading, then the premiums for the higher
issue ages would become very expensive [besides this for safety it would
have to make the size of the commissions for the agents likewise dependable
on the issue age which as already mentioned could lead to all sorts of
difficulties]. The following table shows the various percentage rates which
would remain for the bank from the loaded premium when the loaded premium
is calculated in the usual method according to the Combined Experience
Mortality Table and 3 1/2 percent interest and a loading of 12 1/2 percent.
According to the usual method, the company would lose a ninth part or 11
1/9 percent of the premiums.
| For Administrative Cost Stated as a Percent of the Loaded Premium, if the First Year Premium Is Reduced by: | |||
| Issue Age | Max Rate | 1 % Insured Sum | 1 1/4 % Insured Sum |
|
20
|
8.94%
|
8.04%
|
7.28%
|
|
30
|
8.46%
|
8.55%
|
7.91%
|
|
40
|
7.80%
|
9.01%
|
8.49%
|
|
50
|
7.14%
|
9.41%
|
8.99%
|
|
60
|
6.53%
|
9.71%
|
9.36%
|
This formula does not work for the second and third columns.
Additionally this table shows that the agency commissions which according
to the earlier method were measured according to a determined percentage
rate of the premium, according to the new method could be easily so determined
that it becomes more advantageous for the company. After a significant
commission in the first year after the closing of the insurance policy
the agent will be readily satisfied with a smaller commission for the later
years. If, for example, earlier a company paid as agents commission 6 percent
of the premium, it would have only 5 1/9 percent for the other administrative
costs. Now, if besides the closing commission it pays only 2 to 3 percent
premium commission, there remains for it for most issue ages more than
5 1/9 percent.
As it concerns the reserve for an insurance policy that has existed
for a number of years, the reserve would be somewhat less than according
to the previous method. The difference between the two reserves will become
smaller each year. As can easily be proved with the help of mathematical
symbols, when the calculable rate of the closing commissions is equal to
"a" percent of the insured amount, then the difference in the reserves
is "a" percent of the insured sum lessened by "a" percent of the reserve
calculated according to the old method.
Since the reserve gets bigger every year, the difference will become less with every year. For example, if the old reserve for an insurance policy of $100 has the value of:
Table I (attached) follows the reserve for an insurance policy of $100
for the issue ages of 30,40,and 50 through the whole term of the insurance
policy. We see that the reserves with calculation of closing costs grow
faster than the old reserves as it must be since from the second policy
year on there is a higher, calculated net premium. The Tables II, III,
and IV each follow a company of persons of the same age that insure themselves
at the same time, and Table V gives a picture of the results achieved by
an insurance bank when it reinsures every year such cohorts as are followed
individually in Tables II, III, and IV. It shows that after a sequence
of 60 years, with an insurance inforce of about $41,900,000, the old reserve
would have grown to $12,644,900. The reserve with 1 percent closing costs
would have grown to $12,352,000, and the reserve with 1 1/4 percent closing
costs would have grown to $12,279,100. The difference between the old and
the new reserve is, according to the above, with 1 percent closing costs
$419,102 - $126,449 = $292,653 with 1 1/4 percent closing costs $419,102
- $126,449) * 5/4 = $365,817.
With the assumptions of Table V, a stationary state comes after 60 years, that is, as the consequence of annual cases of death, respectively, expirations of insurance policies, the same sum comes due which is annually newly insured. Therefore, in the future the insurance inforce, premium income, and reserve remain constant as long as the assumptions are met. In such a state, obviously, the net premiums and the calculable interest on those premiums and on the reserve capital must be as great as the annually payable insurance sum. From the existing insurance inforce after 60 years come:
$16,947,500 to members that join at age 30
$20,055,100 to members that join at age 40
$ 4,907,600 to members that join at age 50
If we now add to these values the expected premium and interest
income sums and calculate premium incomes as follows:
| Net Premium if the First Year's Premium is Reduced by | |||
| Issue Age |
|
|
|
|
30
|
1.7973
|
1.8490
|
1.8620
|
|
40
|
2.4841
|
2.5428
|
2.5575
|
|
50
|
3.7083
|
3.7792
|
3.7969
|
ßx = Px + I / äx
We get the following income schedule which should cover the operations
of the bank:
1) when the reserve is calculated without closing costs
Net Premium $1,021,6612) if the reserve is calculated with 1 percent closing costs
3 1/2 percent interest thereon 35,758
3 1/2 percent interest on reserve 442,547
___________
Total $1,499,966
Net Premium (after deduction of $15,000) $1,031,5473) if the reserve is calculated with 1 1/4 percent closing costs
3 1/2 percent interest thereon 36,104
3 1/2 percent interest on the reserve 432,327
-----------
Total $1,499,978
Net Premium (after deduction of $18,750) $1,034,019The small differences of $34, $22, and $23 respectively against the required sum of $1,500,000 have their origin only in the round off that the numbers in Table V were subjected to (this is an alternative proof of the adequateness of the various reserves).
3 1/2 percent interest thereon 36,190
3 1/2 percent interest on the reserve 429,768
------------
Total $1,499,977
The above demonstrates clearly that a reserve calculated with a rate of closing costs that is not too high is completely sufficient for the solvency of the bank if the insurance polices exist until their normal end. If an insured allows his insurance to lapse while he is still alive, then this would not be awkward for the bank especially when it has used the precautionary rule that the negative reserves at the younger issue ages are not put up with the calculable amount but with the amount of zero, and when the agent does not receive a corresponding portion of the closing commissions or must pay out respectively in the case where the insurance lapses during the first year of its existence. On the contrary the collective reserves on such insurance policies go to the bank and flow to its profits. If at lapsing the insurance has been in existence long enough so that the insured can demand a buy back price, only the lower reserve can come into the calculations. As far as we know a bank is (not)obliged in such a case to pay the whole reserve, only a certain portion of the reserve. If a company, for example, customarily pays 3/4 of the reserve, then it can pay back just as much as when the reserve was calculated without closing costs as soon as the reserve has grown, for a $100 insured sum with 1 percent closing costs, to over $3 and, with 1 1/4 percent closing costs, to over $3 3/4 or, when the reserve without closing costs, had grown to $4 or $5, respectively, and it can pay back better since it has already paid itself for all closing costs.
II
Mr. Wilhelm Lazarus in the above mentioned paper "The Security of Life
Insurance Institutions" (Rundschau der Versicherungen, 1857, p.317)
called attention to it, noting that in the closing of the books of the
Gresham Life Insurance Society on 12 November 1855, the liabilities of
the society toward its insureds, even though it had been in existence for
7 years already, were valued lower than the value of premiums yet to be
collected and yet a significant sum of the premiums to be used for the
coverage of obligations had already been collected. A similar relationship
shows itself in the balance sheet of the Great Britain published some time
ago. After an almost 20 year existence, instead of a reserve, the Great
Britain had as an asset the value of future premium income 529,469 pounds
and as a liability the present value of the insured sum 494,707 pounds.
Not only is there no reserve on hand, but the company figures for itself
a net worth or profit of 34,762 pounds or $231,700 which sum is the present
value of what the company will collect in excess of what it expects to
pay at the close of
Note: 34,762 pounds is 231,700 talers. This is 6 2/3 talers per pound.
The exchange rate to U.S. dollars at this time was 5 dollars per pound.
Thus the exchange rate was 1 1/3 taler per dollar or $.75 for a taler.
the existing insurance policies. It is known that in the calculation
of the net premium, the obligations of the insured are equalized with the
obligations of the bank. If we continue on to the reserve calculation with
net premium, then under special circumstances, as was shown in the first
segment, the reserve of a portion of the insurance policies flowing out
of the current year can become negative in calculation but this negative
amount is of little significance as opposed to the reserves of all the
insurance policies of that year and disappearing as opposed to the total
reserves of all insurance policies from various years. It is then impossible,
using calculations with net premiums, for the reserve to work out in the
way it did with the Great Britain; and, in practice, a significant number
of English companies calculate their reserves so that the premiums to be
paid in the future by the insureds are calculated not according to the
net premiums but according to a much higher premium and as a rule according
to the loaded premium while the obligations of the company are valued according
to the net premiums. As a consequence of this calculation, the reserve
does not meet the demand that it be formed out of portion of the premium
demanded for the risk after a one-time deduction of expenses covering calculable
mortality and the addition of calculable interest. This method of calculating
the reserves is clearly misleading. Apparently it is calculated according
to the same principles as in the calculation with net premiums. One deducts
from the value of the insured sum [the net single premium for insurance]
the value of the premiums to be collected in the future, only instead of
the net premium one takes the gross premiums. Herein lies the mistake.
This principle for the calculation of reserves is generally only correct
[even if it can be directly deduced] because and so far as it is a reformulation
of the other principle according to which the reserve consists of the portion
of the net premium not used up for mortality and the duly collected interest.
If a company would enjoy a growing advantage out of closed insurance policy
as soon as possible, then there is only the following way to sanction this
procedure. So long as the loaded premium paid by an insured is higher than
the net premium of that age into which the insured is gradually moving,
the company figures as net premium the premium for one year term insurance,
the excess of the loaded premium over the one year term premium it figures
as profit. Once the insured has reached that age for which as issue age
the net premium of the whole life insurance equals that premium which the
insured pays, the whole life premium is charged. During the first (5 to
7) years when this forms absolutely no reserve, and only later would the
company have a portion of the premium for reserve. Even with this model
which borders on the imaginable, there is no talk about negative reserves.
This is a new proof that this method of the English companies for calculating
reserves with gross premiums is misleading.
The just described way of enjoying the advantage from an insurance policy
as quickly as possible is surpassed by the method using gross premiums.
Here in the moment of the closing of an insurance policy, a profit appears
for as many percent of the total value of the future premium payments as
the percentage loading included in the premiums, or since the value of
the future premium payments measured according to net premiums is equal
to the net single premium for the insurance there is a profit in the amount
of just as much a percent of the net premium, as the loaded premium is
higher than the net premium; for example, if a company figures according
to the Combined Experience Table with an interest rate of 4 1/2 percent
[in many cases, a higher interest rate is figured and the thus reduced
net premiums are loaded on this much more] and 30 percent load of the net
premiums, at the moment of the closing of the insurance policy the profit
amounts to:
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(.30 ) Ax / 100
If such a company in a year closes insurance policies in the amount of $5,000,000 [on the average issue age 37] , in so doing it figures for itself a profit of not less than $480,000, that is,
$5,000,000 * 9.60%.
Therefore the first method cannot be sanctioned because nothing
remains for the company to take care of business expenses if the net premium
to be calculated for obligations of the bank [reserve and mortality] during
the term of the contract reaches the magnitude of the loaded premium .
If the company has significant stock capital, it can cover those costs
with the interest on this capital when the business expansion is small.
If the expansion of the business exceeds a certain limit then the interest
on even the most significant capital will not be sufficient. One cannot
sanction a model which, with the expansion of the business, does not offer
growing means for the coverage of business costs -- with traditionally
figuring banks, the means for the coverage of business costs and beyond
that for the formation of profits, increase first in a determined relationship
to premium income and then in a certain relationship to premium reserve
in that the interest realized above the regular percentage rate [and this
will probably not be higher than 3 1/2 percent in solid companies] flow
not to the reserve but to the administrative accounts, respectively, the
profit. This last income as a rule is completely missing in the irrationally
figuring companies because usually they have a calculated interest rate
that is so high that the actual interest on the reserve capital is actually
much lower than that required by calculation.
The method of reserve calculation form the gross premiums suffers from
the same problem, that it leaves nothing left for coverage of administrative
costs from the premiums that will come in the future. It brings with it
several disadvantages, that bring with them a soon and sure ruin of the
company.
First we will compare the magnitude of the reserves that have been calculated according to the method of gross premiums with the magnitude of rationally calculated reserves. Table I shows in its fourth column the reserve for issue ages 30, 40, and 50 on the basis of a premium provided with an interest rate of 4 1/2 percent and a loading of 30 percent. The table shows that in the first 5 to 7 years, the reserve is negative. Later the reserve achieves a positive value and when the insurance policy continues until its end, the reserve finally grows to the full amount of the insurance sum. Table II shows in its seventh column the reserves which the bank puts up for the number of persons who at issue time had been 30 years old and insured altogether for $500,000 when the appropriate insurance policies are followed up to the natural end. Likewise Table III and IV show similar results for people who at issue age are 40 respectively 50 years old. Columns 8 and 9 in Tables II through IV also demonstrate the asset and liability postings that are so beloved in the English accounting. The difference of the numbers from columns 8 and 9 give the reserves under column 7. Table V contains the summary of the three preceding tables. When we add numbers of the individual columns from the top down, we get those results that an insurance company achieves when it insures such groups of insurance policies each year. The row sums combined out of the 3 tables gives then the numbers for Table V. The premium rates [Column 3] that were the basis for the premium income are the premiums of 4 1/2 percent raised by 30 percent. The columns 8 and 9 were omitted but they could be developed by the addition of the 3 Tables II to IV.
Table V shows that the total reserve becomes positive only in the 13th
year, and that finally after 60 years, where under the assumed circumstances,
likewise takes place a stationary condition, the reserve is about $4,000,000
smaller than the reserve found by rational calculation in columns 4, 5,
and 6. This difference amounts to about 10 percent of the insurance sum,
more than 30 percent of the rational reserve, and about 350 percent of
the premium income. These relationships become even more colossal if we
presume a steadily growing business as it happens in Table IV. Here after
30 years the difference between the reserves from the gross premium and
the rationally calculated reserve already amounts to about $9,000,000,
that is, more than 50 percent of the rational reserve.
To examine further what influence the calculation with gross premiums
has on the development of the business, we will first assume that no insurance
policy expires before its regular end. If the company does fora longer
time equal business then, and this is certainly an unnatural phenomenon,
the profit lessens from year to year. If, for example, we use the numbers
of Table V and assume the existence of a fund from which the company can
withdraw the artificially calculated profit for as long as the actually
collected premium moneys do not reach so far and if we further assume that
all funds draw interest at 4 1/2 percent, and if we calculate agency commissions
as 1/2 of the first annual premium and only 1 percent of all later premiums
and as administrative costs annually $20,000 multiplied by .1 percent of
the insurance sum, the company will achieve profits as they are put together
in the following table:
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1
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$120,000
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$43,800
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274%
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5
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$111,400
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$213,900
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52%
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10
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$101,700
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$413,800
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25%
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15
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$92,200
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$597,300
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15%
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20
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$83,400
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$760,700
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11%
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30
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$67,600
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$1,012,400
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7%
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60
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$50,500
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$1,209,500
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4%
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(table from page 28)
Profit therefore decreases by about 58 percent in the course of time
since with an equal or even business, the profit becomes less each year.
It is a necessary consequence that the whole profit that a policy can bring
with regular existence will be used up in the year of the end of the insurance
policy. If the business continues in the same way, there would result year
after year the same profit if the business expense remains the same. But
these business expenses grow with the expansion of the business; therefore
the profit must decline. Conversely, if a company wants to achieve equal
profits in each following year, it must do more and more business so that
the profit will not sink under a certain percent of the premium income
[In the last case, the profit from a certain point in time on would have
to grow proportionately to the premium income].
This appearance which characterizes the whole system as unhealthy emerges
even without lapses. But what happens when a large part of the insurance
policies lapse time and again? Then here emerges the main problem of this
system. The lapsing of insurance policies occurs, to a large extent, in
insurance policies that are not very old yet, where as a rule the reserve
is still negative. With the lapsing of such an insurance policy, the negative
reserve, respectively, the excessive asset posting disappear; the company
now must cover this disappearance by new insurance policies which bring
with them negative reserves and only with growth beyond that comes the
attainment of the annual profit. Since with growing expansion, the number
of lapsing insurance policies also grows then the business has to grow
faster and faster. Since the accomplishment of this demand requires superhuman
efforts, the sad consequences of this unhealthy system are very soon demonstrated
here too. Even with the calculation with gross premiums according to which
the reserve, the debt of the bank to its insureds, is established so small
in the balance sheet even sometimes conversely as a debt of the insured,
the companies very soon get into difficulties because the expansion is
not increasing sufficiently [the losses in interest income because of the
too high assumed interest rate are not even calculated here]. To figure
out the proper profit, the to cover the no longer manageable deficit, the
companies are forced to extrapolate all kinds of doubtful postings in accruals,
and it is very difficult for the public to judge their value. When this
manuever finally will not work any more, one attempts to get rid of the
whole business by ceding it to another company. It would hardly seem possible
that a company would get rid of a business in such a sickly state; but
the experience shows that every year in England so and so many amalgamations
and business transfers occur. How a company can feel motivated to take
over such an unhealthy business cannot even be conceived in a German imagination.
Such companies can only speculate that in the next years the premium income
will exceed the expenditures for mortality and that they could in some
one form calculate the excess part of the premium income as gain. When
it finally will not go any farther, the business collapses; it does no
harm when the many insureds remain unsatisfied. The executives of the business
and their shareholders, these too have enjoyed their benefits. Nothing
keeps them from establishing a new business under some other name. A fraternal
burial society which is arranged as badly as possible, which collects contributions
according to the principle of the present need, is to be preferred to such
a life insurance company. When the burial society cannot continue on the
whole, the members have paid as little as possible, and certainly too little;
however, the policy holders of such a company have paid sufficient amounts
and yet come out empty in the end.
When an insured, who has already been insured long enough so that for
him, individual calculations result in a positive reserve, wants to give
up his insurance policy, the company according to calculations could concede
a buy back price, but in reality there is no money, for against the positive
reserve of the long term insurance falls the negative reserve of a younger
insured, and both reserves together cancel each other partly or fully.
Therefore, as a rule, one seeks vainly in the insurance terms of such companies
for a rule which obliges the company to buy back the policy; many companies
declare frequently, naturally without adding the true reason, that a buy
back in mass would be disadvantageous to the company. Naturally when one
has to pay more than one has, that can hurt.
We can establish different consequences of the unhealthy system of calculating
with gross premiums. We want to mention only the following. When a company
suddenly ceases every entry of new insurance policies, in the rational
calculation the reserve will grow for a time if a stationary state had
not already been entered. In any case the reserve grows less than when
the new insurance policies are added. It is reversed in a system of gross
premiums. If insurance policies are not being added any more during a series
of years, the reserve will grow stronger than when new insurance policies
are added.
Several companies, that evidently are calculating with gross premiums,
declare that they in the calculations of the value of future premium payments
set apart a sufficient portion of the loaded premiums for the administrative
costs. Suppose now that the value of the insured sums is calculated not
only according to the same mortality table, but also the same interest
rate as the value of the future premium payments [a higher interest rate
in the calculation of the value of the insured sums, and many companies
calculate this way, obviously would form this value as less and thereby
cancel the setting aside of a part of the loaded premium for the administrative
costs when not leading to worse results]; the partitioning of the whole
loading would lead to a rational establishment of the reserve. As a rule,
the premiums have colossal loadings, occasionally 40% or more of the net
premiums and above; if now about 5 percent of it is partitioned, then the
premiums that are to be used for the calculation of future premium income
are loaded so sufficiently that the reserve for the individual insurance
policy here also needs several years to go over from the negative to the
positive. If the annual business profit with equal additions and without
losses by lapsing would become equal [the same year after year], that is,
in the sum, not in the percent of premium income then since a portion of
the insurance policies will lapse to achieve the equal gain, the annual
additions must grow or increase, and since with growing additions there
also grows the number of lapsing insurance policies, the additions must
grow for so much more. Should the annual profit not sink below a certain
percentage rate of the premium income, the annual additions must naturally
grow so much faster. Here as above, the condition of the company is tied
to the demand that the annual additions must grow faster each year. Also
for this case, there result the same consequences as when the reserves
are calculated according to the loaded premiums only here, they occur somewhat
slower and milder.
If we summarize the above, then the mistakes of the reserve calculation from gross premiums which are higher than the net premiums calculated for the risk, but in general equal to the loaded premiums, consists of the following:
1) At the close of an insurance policy, there is calculated an artificial gain which consequently may not be realized and in so and so many cases is not actually realized;
2) The reserve of the individual insurance policy is in and for itself already too low besides that the negative reserves of the younger insurance policies partly or wholly cancel the positive reserves of the older insurance policies;
3) The gain mentioned under 1) is the sole or main source
a) for absorbing the administrative costs,
b) for the coverage of the disappearing negative reserves as consequence of the premature expiration of the insurance policies,
c) for the annual profit.
Since both the first postings [ a) and b)] are necessary expenses and in addition growing ones, but since the profit mentioned under 1) cannot always grow further or what amounts to the same thing since the annual addition or growth cannot go higher forever with each following year, then the annual profit mentioned under c) sooner or later must let up and finally a deficit will come into existence and the bank will become insolvent.
Reserves for an endowment at age 90 with a face amount of $100 and premiums
payable annually, calculated according to the Combined Experience Table.
Table I. Zillmer Reserve Calculations Endowment @90 Interest Rate 3.5% Issue Age 30 Net Premium 1.797252 I= 0 I= 1 I= 1.25 ----------------| ----------------| ----------------| Initial Terminal Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves Reserves Reserves ------------------------------------------------------ 1 1.7973 1.0263 0.7973 0.0366 0.5473 -0.2109 2 2.8236 2.0824 1.8338 1.1033 1.5864 0.8585 3 3.8797 3.1685 2.9005 2.2002 2.6557 1.9581 4 4.9658 4.2859 3.9974 3.3287 3.7554 3.0894 5 6.0831 5.4359 5.1260 4.4903 4.8867 4.2539 6 7.2332 6.6190 6.2875 5.6852 6.0511 5.4518 7 8.4163 7.8367 7.4825 6.9151 7.2490 6.6847 8 9.6340 9.0905 8.7123 8.1814 8.4819 7.9541 9 10.8878 10.3811 9.9787 9.4849 9.7514 9.2608 10 12.1783 11.7101 11.2821 10.8272 11.0581 10.6065 11 13.5073 13.0794 12.6244 12.2102 12.4037 11.9929 12 14.8767 14.4899 14.0075 13.6348 13.7902 13.4211 13 16.2872 15.9415 15.4321 15.1009 15.2183 14.8907 14 17.7387 17.4306 16.8981 16.6049 16.6880 16.3985 15 19.2279 18.9528 18.4022 18.1423 18.1958 17.9397 16 20.7501 20.5055 19.9396 19.7106 19.7370 19.5118 17 22.3028 22.0830 21.5078 21.3038 21.3091 21.1090 18 23.8802 23.6846 23.1011 22.9214 22.9063 22.7307 19 25.4818 25.3086 24.7187 24.5617 24.5279 24.3750 20 27.1059 26.9545 26.3590 26.2240 26.1723 26.0414 21 28.7517 28.6203 28.0213 27.9065 27.8387 27.7281 22 30.4176 30.3045 29.7038 29.6075 29.5254 29.4333 23 32.1017 32.0050 31.4048 31.3250 31.2305 31.1550 24 33.8022 33.7198 33.1223 33.0570 32.9523 32.8913 25 35.5171 35.4489 34.8543 34.8034 34.6886 34.6420 26 37.2462 37.1891 36.6007 36.5609 36.4393 36.4039 27 38.9863 38.9387 38.3582 38.3281 38.2012 38.1754 28 40.7360 40.6982 40.1253 40.1052 39.9727 39.9569 29 42.4954 42.4647 41.9024 41.8893 41.7542 41.7455 30 44.2619 44.2360 43.6866 43.6783 43.5427 43.5389 Issue Age 30 (Continued) I= 0 I= 1 I= 1.25 ----------------| ----------------| ----------------| Initial Terminal Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves Reserves Reserves ----------------------------------------------------- 31 46.0332 46.0064 45.4756 45.4665 45.3362 45.3315 32 47.8037 47.7736 47.2637 47.2513 47.1287 47.1208 33 49.5709 49.5334 49.0486 49.0288 48.9180 48.9026 34 51.3307 51.2838 50.8260 50.7967 50.6998 50.6749 35 53.0811 53.0210 52.5939 52.5512 52.4721 52.4338 36 54.8182 54.7418 54.3485 54.2892 54.2310 54.1761 37 56.5391 56.4441 56.0865 56.0085 55.9733 55.8996 38 58.2413 58.1243 57.8058 57.7055 57.6969 57.6008 39 59.9215 59.7814 59.5028 59.3792 59.3981 59.2787 40 61.5787 61.4155 61.1765 61.0296 61.0759 60.9332 41 63.2127 63.0242 62.8269 62.6545 62.7304 62.5620 42 64.8215 64.6072 64.4517 64.2532 64.3593 64.1647 43 66.4044 66.1636 66.0505 65.8252 65.9620 65.7406 44 67.9608 67.6942 67.6225 67.3711 67.5379 67.2903 45 69.4914 69.1987 69.1684 68.8907 69.0876 68.8137 46 70.9959 70.6789 70.6879 70.3857 70.6109 70.3124 47 72.4761 72.1380 72.1829 71.8594 72.1096 71.7897 48 73.9353 73.5777 73.6566 73.3135 73.5870 73.2474 49 75.3750 75.0023 75.1107 74.7523 75.0447 74.6898 50 76.7995 76.4206 76.5495 76.1848 76.4870 76.1259 51 78.2179 77.8448 77.9821 77.6232 77.9232 77.5678 52 79.6420 79.2939 79.4205 79.0868 79.3651 79.0350 53 81.0911 80.7952 80.8841 80.6032 80.8323 80.5551 54 82.5925 82.3844 82.4004 82.2082 82.3524 82.1642 55 84.1816 84.1150 84.0055 83.9561 83.9615 83.9164 56 85.9122 86.0601 85.7534 85.9207 85.7136 85.8859 57 87.8574 88.3378 87.7180 88.2212 87.6831 88.1920 58 90.1350 91.1447 90.0184 91.0562 89.9893 91.0340 59 92.9420 94.8211 92.8534 94.7693 92.8313 94.7564 60 96.6184 0.0000 96.5666 0.0000 96.5536 0.0000 Table I. Zillmer Reserve Calculations Endowment @90, Load 30% NSP Interest Rate 4.5% Issue Age 30 Net Premium 1.609444 I= 0 I= 8.1619 ----------------| ----------------| Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves ------------------------------------ 1 1.6094 0.8465 -6.5525 -7.2463 2 2.4560 1.7234 -5.6369 -6.2978 3 3.3329 2.6312 -4.6884 -5.3160 4 4.2406 3.5714 -3.7065 -4.2990 5 5.1808 4.5458 -2.6896 -3.2451 6 6.1552 5.5550 -1.6356 -2.1535 7 7.1645 6.6010 -0.5440 -1.0221 8 8.2104 7.6857 0.5873 0.1511 9 9.2951 8.8101 1.7605 1.3672 10 10.4195 9.9763 2.9767 2.6287 11 11.5858 11.1869 4.2381 3.9380 12 12.7963 12.4430 5.5475 5.2967 13 14.0525 13.7451 6.9061 6.7051 14 15.3546 15.0902 8.3145 8.1600 15 16.6997 16.4741 9.7694 9.6568 16 18.0836 17.8947 11.2663 11.1933 17 19.5041 19.3463 12.8027 12.7634 18 20.9557 20.8287 14.3728 14.3668 19 22.4381 22.3404 15.9762 16.0019 20 23.9499 23.8812 17.6114 17.6685 21 25.4906 25.4495 19.2779 19.3648 22 27.0589 27.0437 20.9742 21.0891 23 28.6532 28.6623 22.6986 22.8398 24 30.2717 30.3032 24.4492 24.6147 25 31.9127 31.9668 26.2241 26.4140 26 33.5762 33.6497 28.0234 28.2343 27 35.2592 35.3507 29.8437 30.0741 28 36.9602 37.0703 31.6836 31.9341 29 38.6798 38.8057 33.5435 33.8111 30 40.4152 40.5547 35.4206 35.7029 31 42.1642 42.3115 37.3123 37.6030 32 43.9210 44.0735 39.2125 39.5089 33 45.6830 45.8365 41.1183 41.4157 34 47.4459 47.5981 43.0252 43.3211 35 49.2076 49.3543 44.9306 45.2206 36 50.9637 51.1015 46.8301 47.1105 37 52.7110 52.8374 48.7199 48.9880 38 54.4468 54.5578 50.5975 50.8489 39 56.1673 56.2617 52.4583 52.6918 40 57.8711 57.9486 54.3012 54.5164 Endowment @90, Load 30% NSP Interest Rate 4.5% Issue Age 30 (Continued) Net Premium 1.609444 I= 0 I= 8.1619 ----------------| ----------------| Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves ------------------------------------ 41 59.5580 59.6158 56.1258 56.3197 42 61.2253 61.2627 57.9292 58.1010 43 62.8721 62.8881 59.7104 59.8591 44 64.4976 64.4925 61.4685 61.5944 45 66.1019 66.0753 63.2038 63.3064 46 67.6847 67.6380 64.9158 64.9967 47 69.2474 69.1840 66.6061 66.6688 48 70.7935 70.7148 68.2783 68.3245 49 72.3242 72.2346 69.9340 69.9684 50 73.8440 73.7533 71.5779 71.6110 51 75.3627 75.2837 73.2205 73.2664 52 76.8931 76.8472 74.8758 74.9575 53 78.4566 78.4743 76.5669 76.7174 54 80.0837 80.2054 78.3268 78.5898 55 81.8148 82.1016 80.1992 80.6407 56 83.7110 84.2472 82.2502 82.9614 57 85.8566 86.7787 84.5709 85.6996 58 88.3881 89.9251 87.3090 89.1028 59 91.5346 94.0843 90.7122 93.6015 60 95.6938 0.0000 95.2109 0.0000 Table I. Zillmer Reserve Calculations Endowment @90 Interest Rate 3.5% Issue Age 40 Net Premium 2.48414 I = 0 I = 1 I = 1.25 ----------------| ----------------| ----------------| Initial Terminal Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves Reserves Reserves ------------------------------------------------------ 1 1.7973 1.5510 0.7973 0.5665 0.5473 0.3203 2 3.3482 3.1486 2.3637 2.1800 2.1176 1.9379 3 4.9458 4.7926 3.9773 3.8405 3.7352 3.6025 4 6.5899 6.4793 5.6378 5.5441 5.3998 5.3103 5 8.2765 8.2033 7.3413 7.2854 7.1075 7.0559 6 10.0006 9.9620 9.0826 9.0616 8.8531 8.8365 7 11.7593 11.7487 10.8589 10.8662 10.6338 10.6456 8 13.5460 13.5627 12.6634 12.6984 12.4428 12.4823 9 15.3600 15.4022 14.4956 14.5562 14.2795 14.3447 10 17.1994 17.2663 16.3535 16.4390 16.1420 16.2321 11 19.0635 19.1531 18.2362 18.3446 18.0294 18.1425 12 20.9504 21.0606 20.1419 20.2712 19.9398 20.0739 13 22.8579 22.9866 22.0685 22.2165 21.8711 22.0240 14 24.7839 24.9290 24.0138 24.1782 23.8212 23.9906 15 26.7262 26.8874 25.9755 26.1563 25.7878 25.9735 16 28.6846 28.8583 27.9535 28.1469 27.7707 27.9690 17 30.6556 30.8400 29.9441 30.1484 29.7663 29.9755 18 32.6373 32.8329 31.9457 32.1612 31.7728 31.9933 19 34.6301 34.8336 33.9584 34.1820 33.7905 34.0191 20 36.6309 36.8399 35.9792 36.2083 35.8163 36.0504 21 38.6371 38.8451 38.0055 38.2336 37.8476 38.0807 22 40.6424 40.8467 40.0308 40.2552 39.8780 40.1073 23 42.6440 42.8399 42.0524 42.2683 41.9045 42.1254 24 44.6372 44.8225 44.0656 44.2707 43.9227 44.1328 25 46.6198 46.7901 46.0680 46.2580 45.9301 46.1249 26 48.5873 48.7391 48.0552 48.2265 47.9222 48.0984 27 50.5364 50.6672 50.0238 50.1738 49.8956 50.0505 28 52.4644 52.5702 51.9711 52.0959 51.8478 51.9773 29 54.3674 54.4471 53.8932 53.9916 53.7746 53.8777 30 56.2444 56.2979 55.7889 55.8609 55.6750 55.7517 31 58.0952 58.1201 57.6582 57.7013 57.5489 57.5966 32 59.9173 59.9129 59.4985 59.5121 59.3938 59.4118 33 61.7102 61.6758 61.3093 61.2926 61.2091 61.1968 34 63.4731 63.4094 63.0898 63.0435 62.9940 62.9520 35 65.2066 65.1134 64.8407 64.7646 64.7492 64.6774 36 66.9107 66.7899 66.5618 66.4578 66.4746 66.3748 37 68.5872 68.4426 68.2551 68.1270 68.1721 68.0481 38 70.2399 70.0733 69.9243 69.7740 69.8454 69.6992 39 71.8705 71.6867 71.5713 71.4036 71.4964 71.3328 40 73.4840 73.2933 73.2009 73.0262 73.1301 72.9594 Issue Age 40 (Continued) I = 0 I = 1 I = 1.25 ----------------| ----------------| ----------------| Initial Terminal Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves Reserves Reserves ------------------------------------------------------ 41 75.0905 74.9063 74.8234 74.6554 74.7567 74.5926 42 76.7035 76.5476 76.4526 76.3130 76.3899 76.2544 43 78.3448 78.2480 78.1103 78.0305 78.0517 77.9761 44 80.0453 80.0480 79.8278 79.8485 79.7734 79.7986 45 81.8453 82.0081 81.6457 81.8282 81.5959 81.7832 46 83.8053 84.2113 83.6254 84.0534 83.5804 84.0139 47 86.0085 86.7910 85.8506 86.6589 85.8112 86.6259 48 88.5883 89.9702 88.4562 89.8699 88.4231 89.8448 49 91.7675 94.1342 91.6672 94.0756 91.6421 94.0609 50 95.9315 0.0000 95.8728 0.0000 95.8581 0.0000 Table I. Zillmer Reserve Calculations Endowment @90, Load 30% NSP Interest Rate 4.5% Issue Age 40 Net Premium 2.265013 I= 0 I= 10.3406 ----------------| ----------------| Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves ------------------------------------ 1 2.2650 1.3447 -8.0756 -8.8569 2 3.6097 2.7400 -6.5919 -7.3172 3 5.0050 4.1864 -5.0522 -5.7213 4 6.4514 5.6806 -3.4563 -4.0726 5 7.9456 7.2179 -1.8076 -2.3764 6 9.4829 8.7958 -0.1114 -0.6352 7 11.0608 10.4083 1.6298 1.1440 8 12.6733 12.0550 3.4090 2.9609 9 14.3200 13.7343 5.2259 4.8139 10 15.9993 15.4458 7.0789 6.7024 11 17.7108 17.1879 8.9674 8.6246 12 19.4529 18.9588 10.8896 10.5787 13 21.2238 20.7567 12.8437 12.5625 14 23.0217 22.5795 14.8275 14.5738 15 24.8445 24.4274 16.8388 16.6127 16 26.6924 26.2969 18.8778 18.6755 17 28.5619 28.1864 20.9405 20.7604 18 30.4514 30.0965 23.0254 22.8681 19 32.3615 32.0242 25.1331 24.9951 20 34.2892 33.9671 27.2601 27.1389 21 36.2321 35.9185 29.4039 29.2921 22 38.1835 37.8758 31.5571 31.4518 23 40.1408 39.8341 33.7168 33.6126 24 42.0991 41.7910 35.8776 35.7718 25 44.0560 43.7417 38.0368 37.9243 26 46.0068 45.6827 40.1893 40.0659 27 47.9477 47.6109 42.3309 42.1935 28 49.8759 49.5220 44.4585 44.3023 29 51.7870 51.4146 46.5673 46.3906 30 53.6796 53.2885 48.6556 48.4582 31 55.5535 55.1405 50.7232 50.5018 32 57.4055 56.9699 52.7668 52.5203 33 59.2349 58.7754 54.7853 54.5126 34 61.0404 60.5576 56.7776 56.4790 35 62.8226 62.3158 58.7440 58.4190 36 64.5808 64.0517 60.6840 60.3344 37 66.3167 65.7690 62.5994 62.2293 38 68.0340 67.4694 64.4943 64.1055 39 69.7344 69.1577 66.3705 65.9684 40 71.4227 70.8446 68.2334 67.8298 Table I. Zillmer Reserve Calculations Endowment @90, Load 30% NSP Interest Rate 4.5% Issue Age 40 (Continued) Net Premium 2.265013 I= 0 I= 10.3406 ----------------| ----------------| Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves ------------------------------------ 41 73.1096 72.5447 70.0948 69.7056 42 74.8097 74.2814 71.9706 71.6219 43 76.5464 76.0888 73.8869 73.6163 44 78.3538 78.0118 75.8813 75.7381 45 80.2768 80.1181 78.0031 78.0622 46 82.3831 82.5015 80.3272 80.6920 47 84.7665 85.3135 82.9570 83.7948 48 87.5785 88.8086 86.0598 87.6514 49 91.0736 93.4288 89.9164 92.7493 50 95.6938 0.0000 95.0143 0.0000 Table I. Zillmer Reserve Calculations Endowment @90 Interest Rate 3.5% Issue Age 50 Net Premium 3.708312 I = 0 I = 1 I = 1.25 ----------------| ----------------| ----------------| Initial Terminal Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves Reserves Reserves ------------------------------------------------------ 1 3.7083 2.2806 2.7083 1.2806 2.4583 1.0591 2 5.9889 4.5862 5.0117 3.6098 4.7674 3.3935 3 8.2945 6.9142 7.3404 5.9616 7.1018 5.7506 4 10.6225 9.2618 9.6916 8.3333 9.4589 8.1276 5 12.9701 11.6290 12.0628 10.7247 11.8359 10.5244 6 15.3373 14.0112 14.4536 13.1313 14.2327 12.9364 7 17.7195 16.4065 16.8597 15.5511 16.6447 15.3616 8 20.1148 18.8153 19.2789 17.9845 19.0699 17.8005 9 22.5236 21.2336 21.7117 20.4276 21.5088 20.2490 10 24.9419 23.6585 24.1543 22.8773 23.9573 22.7042 11 27.3668 26.0823 26.6034 25.3259 26.4126 25.1583 12 29.7906 28.5016 29.0514 27.7699 28.8666 27.6079 13 32.2099 30.9108 31.4949 30.2038 31.3162 30.0472 14 34.6191 33.3071 33.9282 32.6246 33.7555 32.4735 15 37.0155 35.6853 36.3485 35.0272 36.1818 34.8814 16 39.3936 38.0411 38.7505 37.4071 38.5897 37.2667 17 41.7495 40.3716 41.1299 39.7614 40.9750 39.6262 18 44.0799 42.6717 43.4836 42.0851 43.3345 41.9551 19 46.3800 44.9404 45.8068 44.3769 45.6634 44.2521 20 48.6487 47.1774 48.0981 46.6369 47.9605 46.5172 21 50.8858 49.3798 50.3575 48.8618 50.2255 48.7471 22 53.0882 51.5469 52.5820 51.0510 52.4554 50.9412 23 55.2552 53.6777 54.7706 53.2036 54.6495 53.0986 24 57.3860 55.7730 56.9227 55.3204 56.8069 55.2202 25 59.4813 57.8327 59.0391 57.4012 58.9285 57.3056 26 61.5410 59.8591 61.1194 59.4483 61.0139 59.3573 27 63.5674 61.8567 63.1660 61.4663 63.0657 61.3799 28 65.5650 63.8277 65.1835 63.4575 65.0882 63.3755 29 67.5360 65.7779 67.1742 65.4276 67.0838 65.3501 30 69.4862 67.7196 69.1439 67.3893 69.0584 67.3161 31 71.4280 69.6693 71.1052 69.3589 71.0245 69.2902 32 73.3776 71.6531 73.0743 71.3630 72.9985 71.2988 33 75.3614 73.7085 75.0780 73.4394 75.0071 73.3798 34 77.4168 75.8841 77.1539 75.6373 77.0881 75.5826 35 79.5924 78.2532 79.3512 78.0307 79.2909 77.9814 36 81.9615 80.9162 81.7441 80.7209 81.6897 80.6777 37 84.6245 84.0343 84.4337 83.8709 84.3860 83.8348 38 87.7426 87.8770 87.5830 87.7530 87.5431 87.7255 39 91.5853 92.9100 91.4641 92.8375 91.4338 92.8214 40 96.6184 0.0000 96.5475 0.0000 96.5297 0.0000 Table I. Zillmer Reserve Calculations Endowment @90, Load 30% NSP Interest Rate 4.5% Issue Age 50 Net Premium 3.4654 I= 0 I= 13.3771 ----------------| ----------------| Initial Terminal Initial Terminal Reserves Reserves Reserves Reserves ------------------------------------ 1 3.4654 2.0603 -9.9117 -11.0412 2 5.5257 4.1548 -7.5758 -8.6666 3 7.6202 6.2811 -5.2012 -6.2558 4 9.7465 8.4369 -2.7904 -3.8116 5 11.9023 10.6223 -0.3462 -1.3338 6 14.0877 12.8333 2.1316 1.1729 7 16.2987 15.0680 4.6383 3.7065 8 18.5334 17.3270 7.1719 6.2678 9 20.7924 19.6069 9.7332 8.8526 10 23.0723 21.9046 12.3180 11.4577 11 25.3700 24.2126 14.9231 14.0744 12 27.6780 26.5274 17.5398 16.6989 13 29.9928 28.8434 20.1643 19.3248 14 32.3088 31.1578 22.7902 21.9487 15 34.6232 33.4649 25.4141 24.5644 16 36.9303 35.7603 28.0298 27.1669 17 39.2257 38.0408 30.6323 29.7524 18 41.5062 40.3010 33.2178 32.3150 19 43.7664 42.5394 35.7804 34.8528 20 46.0048 44.7555 38.3182 37.3654 21 48.2209 46.9459 40.8308 39.8488 22 50.4113 49.1094 43.3142 42.3017 23 52.5748 51.2448 45.7671 44.7228 24 54.7102 53.3525 48.1882 47.1124 25 56.8179 55.4319 50.5778 49.4699 26 58.8973 57.4849 52.9353 51.7976 27 60.9503 59.5159 55.2630 54.1003 28 62.9813 61.5269 57.5657 56.3803 29 64.9923 63.5236 59.8457 58.6441 30 66.9890 65.5187 62.1095 60.9061 31 68.9841 67.5293 64.3715 63.1857 32 70.9947 69.5833 66.6511 65.5144 33 73.0487 71.7209 68.9798 67.9380 34 75.1863 73.9951 71.4034 70.5164 35 77.4605 76.4862 73.9818 73.3407 36 79.9516 79.3049 76.8061 76.5365 37 82.7703 82.6307 80.0019 80.3072 38 86.0961 86.7643 83.7726 84.9937 39 90.2297 92.2284 88.4591 91.1888 40 95.6938 0.0000 94.6542 0.0000 Table II Overview of the business outcomes which result when a life insurance bank insures a cohort of 30 year olds, each one for the same sum, in a total of $500,000. The annual premium is percent. Life Reserves on the Basis Business Insurance Premium 3 1/2 % Interest & Year Inforce Claims Income I = 0% I = 1% I=11/4% - --- ---- --- ---- --- ---- --- ---- --- ---- --- ---- --- --- 0 500000 1 495800 4200 10458 5100 181 -1045 2 491500 4300 10370 10200 5422 4219 3 487200 4300 10281 15400 10719 9540 4 482900 4300 10191 20700 16074 14919 5 478500 4400 10101 26000 21486 20355 6 474100 4400 10009 31400 26954 25847 7 469600 4500 9917 36800 32473 31391 8 465100 4500 9822 42300 38052 36995 9 460500 4600 9728 47800 43678 42646 10 455800 4700 9632 53400 49350 48344 11 451100 4700 9534 59000 55080 54100 12 446300 4800 9436 64700 60852 59898 13 441400 4900 9335 70400 66655 65728 14 436400 5000 9233 76100 72464 71563 15 431300 5100 9128 81700 78248 77374 16 426000 5300 9021 87400 83967 83120 17 420500 5500 8910 92900 89583 88763 18 414800 5700 8795 98200 95078 94287 19 408900 5900 8676 103500 100433 99669 20 402700 6200 8553 108500 105604 104869 21 396300 6400 8423 113400 110594 109886 22 389600 6700 8289 118100 115351 114672 23 382600 7000 8149 122500 119850 119199 24 375300 7300 8003 126600 124063 123441 25 367700 7600 7850 130300 127972 127379 26 359700 8000 7691 133800 131510 130945 27 351400 8300 7524 136800 134685 134149 28 342700 8700 7350 139500 137440 136932 29 333700 9000 7168 141700 139785 139305 30 324300 9400 6980 143500 141649 141197 31 314500 9800 6783 144700 142992 142568 32 304200 10300 6578 145300 143739 143341 33 293500 10700 6363 145400 143899 143529 34 282400 11100 6139 144800 143450 143106 35 270900 11500 5907 143600 142361 142043 36 259000 11900 5666 141800 140609 140316 37 246700 12300 5417 139200 138173 137904 38 234000 12700 5160 136000 135031 134786 39 221000 13000 4894 132100 131228 131006 40 207700 13300 4623 127600 126759 126558 Table II (Continued) Life Reserves on the Basis Business Insurance Premium 3 1/2 % Interest & Year Inforce Claims Income I = 0% I = 1% I=11/4% - --- ---- --- ---- --- ---- --- ---- --- ---- --- ---- --- --- 41 194200 13500 4344 122400 121675 121495 42 180600 13600 4062 116700 116041 115882 43 166900 13700 3778 110400 109862 109721 44 153200 13700 3491 103700 103213 103089 45 139600 13600 3204 96600 96171 96064 46 126300 13300 2920 89300 88897 88805 47 113300 13000 2642 81700 81417 81338 48 100700 12600 2370 74100 73827 73760 49 88600 12100 2106 66500 66231 66175 50 77100 11500 1853 58900 58739 58693 51 66300 10800 1613 51600 51464 51427 52 56300 10000 1387 44600 44526 44497 53 47100 9200 1178 38100 37964 37941 54 38800 8300 985 32000 31897 31880 55 31400 7400 812 26400 26362 26350 56 25000 6400 657 21500 21480 21471 57 19400 5600 523 17100 17115 17109 58 14700 4700 406 13400 13385 13382 59 10800 3900 307 10200 10235 10234 60 0 10800 226 0 0 0 Table II (Continued) Age 30 Reserves Asset Liability Business Insurance Premium 4 1/2% NPV NPV Year Inforce Claims Income 30% Load Premiums Benefits 0 500000 1 495800 4200 10458 -35900 173800 137900 2 491500 4300 10370 -31000 170800 139900 3 487200 4300 10281 -25900 167700 141900 4 482900 4300 10191 -20800 164600 143900 5 478500 4400 10101 -15500 161500 146000 6 474100 4400 10009 -10200 158300 148200 7 469600 4500 9917 -4800 155100 150300 8 465100 4500 9822 700 151800 152600 9 460500 4600 9728 6300 148500 154800 10 455800 4700 9632 12000 145100 157100 11 451100 4700 9534 17800 141700 159500 12 446300 4800 9436 23600 138200 161800 13 441400 4900 9335 29600 134600 164300 14 436400 5000 9233 35600 131000 166700 15 431300 5100 9128 41600 127400 169100 16 426000 5300 9021 47700 123700 171400 17 420500 5500 8910 53700 119900 173600 18 414800 5700 8795 59600 116100 175700 19 408900 5900 8676 65400 112300 177700 20 402700 6200 8553 71200 108400 179600 21 396300 6400 8423 76700 104500 181200 22 389600 6700 8289 82200 100500 182700 23 382600 7000 8149 87400 96500 183900 24 375300 7300 8003 92400 92500 184900 25 367700 7600 7850 97100 88500 185600 26 359700 8000 7691 101600 84400 186000 27 351400 8300 7524 105700 80300 186000 28 342700 8700 7350 109400 76300 185700 29 333700 9000 7168 112800 72200 185100 30 324300 9400 6980 115800 68200 184000 31 314500 9800 6783 118300 64200 182400 32 304200 10300 6578 120200 60200 180400 33 293500 10700 6363 121600 56200 177800 34 282400 11100 6139 122300 52300 174700 35 270900 11500 5907 122500 48500 171000 36 259000 11900 5666 122000 44800 166800 37 246700 12300 5417 120900 41100 162000 38 234000 12700 5160 119000 37600 156600 39 221000 13000 4894 116400 34200 150600 40 207700 13300 4623 113200 30900 144100 41 194200 13500 4344 109400 27700 137100 42 180600 13600 4062 104900 24700&nbs