APPENDIX.

APPENDIX.

The following tables are valuable to the managers of the Fraternal Orders as well as to actuaries. The commutation columns are given that the latter may use them for original work, or for verifying statements and computations in the body of this book, if so desired. The Annuities, Single and Level Premiums, and Reserve Values are given for the use of officials.

At this time, when changes are contemplated, good use may be made of the columns of the Annuities and Single Premiums, and it may be of advantage to indicate the manner of use.

It has been explained, under the headings of "Cost of Protection" and "Valuation" in the body of the book, that the Single Premium represents the present value of the sum insured, or the present value of the promised benefits, or the aggregate of the present value of all of the yearly costs of insurance, or the ultimate value of the certificate. All of these are synonymous expressions. The column of " Single Premiums for $1,000, Whole Life Insurance,” in the following Table, gives, therefore, the present value of a benefit of $1,000 payable at death, and also represents the amount necessary to be in hand, in one lump sum, to pay for $1,000 of insurance protection through the whole of life; that is, it represents the sum of all of the yearly costs of insurance during the life of the insured. Thus, at age 20 the present value of a life insurance benefit of $1,000 is $211.86; at age 30, $266.23; at age 40, $343.31; at age 50, $445.53; at age 60, $570.70, etc.

The column of Life Annuities represents the single premium," or amount of one lump sum necessary to pay $1.00 each year to the annuitant during his life time. Thus, it will require $20.49157 in hand to pay $1.00 annually for life to a man aged 20; and $19.07797 for a life annuity of $1.00 to a man aged 30; $17.07395 for a man at 40; $14.41623 at 50; and 11.16191 for a $1.00 annuity to a man aged 60.

By dividing the Single Premium, at any age, by the Life Annuity of $1.00, at the same age, the net annual Level Premium for $1,000 of whole life insurance is obtained. And, of course, by multiplying the Life Annuity by the Level Premium, at any age, the Single Premium is the result.

Taking the Level Premium at any age, say 35, $16.615, and multiplying by the Annuity, $18.1565, will not only give the Single Premium, $301.67, but it also gives the present value of all the future premiums, (including the first) of $16.615 each, which are expected to be received during the life time of the man entering upon his insurance at age 35.

By multiplying the same Level Premium, $16.615, by the Annuity at age 36, $17.9532, the present value is obtained of all the future premiums receivable, beginning with the second year of insurance, from the man entering at age 35. This amount is found to be $298.29, while the present value at beginning of the first year was $301.67. The present value has decreased, at beginning of second year, because one premium payment has been made. By multiplying the same Level Premium ($16.615) by the Annuity at age 37 ($17.7432) the present value of the future premiums ($294.80) from beginning of the third year of insurance can be obtained. This amount ($294.80) is less than the amount at the beginning of the second year ($298.29) because two premium payments have been made and used for insurance purposes. After the payment and use of each annual premium, the present value of the future receivable premiums becomes less, the amount of which may be determined, as above, at the beginning of any insurance year. It is in this way that the present value of the premiums receivable are determined when a valuation is made. By subtracting the present value of the

future Level Premiums from the Single Premium (or present value of the benefit assured) the Reserve is obtainable—that is, the amount of necessary accumulation to be in hand to make the present value of the premiums receivable equal to the present value of the benefits promised. (See under "Valuation" in body of book relating to "Fraternal Beneficiary Orders"-also under Chief Registrar on Valuation "in Part I relating to "English Friendly Societies.") At the beginning of the first year the present value of the benefit and the present value of the future premiums (including the first payment) are the same, $301.67. At beginning of second year the present value of the benefit ($309.49) is $11.20 greater than the present value of the future premiums ($298.29), and requires the reserve accumulation of $11.20 to keep the values in balance. The difference between the two present values increases with each year of the insurance and a valuation of the certificates discovers the amount of the reserve necessary to maintain the premium level and uniform for life. The differences between the present values of benefits and the present values of future premiums, for each age of entry from 20 to 70 inclusive, and for each insurance year from the age at entry to the end of the Mortality Table (age 99), have been worked out and will be found in the table of Terminal Reserve Values.

The Annuities, Single Premiums, Level Premiums and Reserve Values are the results of calculations based upon the actual experience of Fraternal Orders and cannot be ignored by officials who have regard for the future welfare of their societies. These tables should be used to test the rates of assessment fixed by them, and to test the sufficiency of emergency or reserve funds which they are accumulating.

For instance, take the yearly rates of two societies, both of which are represented as high enough to remain level throughout life, and being, at age 35, $15.73 (expense percentage deducted) and $11.76 respectively. Conceding that the Level Rates by the National Fraternal Congress Table are not too high (a fact that can be demonstrated from the experience of these two Orders), it appears that one of the societies lacks 82 cents and the other $4.86 of having an adequate premium to provide for the $1000 death benefit promised. To appreciate the full significance of these deficiencies, get the present value by multiplying each by the Life Annuity at age 35, $18.1565. It is found that the premium deficiency of 82 cents is equal to a present sum of $14.89, and the premium deficiency of $4.86 is equal to a present sum of $88.24. That is to say, in order to make the rate of $15.73 adequate and sufficient for a whole life level rate to assure a death benefit of $1000 it would be necessary to supplement it by a present sum of $14.89 in hand, at the beginning of the insurance contract. To make an adequate level rate of $11.76 for the benefit promised would require a present sum in hand of $88.24. These amounts, improved at compound interest, will give the increasing values of the premium deficiencies for the future years of the insurance contract. This point is dwelt upon by the Actuaries of the Manchester Unity and Ancient Order of Foresters in their reports, quoted in the review of those societies. (Ante.)

When rates are too low, each new member brings a premium deficiency on entering the Order, and the sum of such deficiencies will ultimately and surely result in hopeless insolvency. The present value of such deficiencies, now existing in one of the prominent Orders, is given in previous pages, being calculated for each age, and showing the alarming aggregate of more than fifty millions of dollars, on the assumption that members are paying the regular rates at their attained ages. The fact that they are paying at ages of entry will swell the total deficiency to nearly eighty millions.

If properly used, the tables given herewith will be of great value to officials of Fraternal Orders.

The following are the Level and Step-Rates and modifications recommended by the National Fraternal Congress to its members, and urged upon the Convention of Insurance Commissioners (in 190v) as minimum rates for new Fraternal Insurance Orders:

LEVEL RATES FROM AGE OF ENTRY.

This table shows the lowest rates that can be deduced from the mortality table above. The full amount must be collected annually and the portion not used to provide for current mortality must be invested at 4 per cent. interest. The annual rate is calculated on the basis that the full amount is paid at the beginning of the year. The monthly rates are increased slightly to provide for the loss of interest due to that method of payment and the slightly less amount contributed by dying members.

STEP-RATE AND TWO EXAMPLES ON MODIFIED STEP-RATES.

Column 1 gives the age of groups. Column 2 gives the annual rates for the natural step-rate to age 61 and level rate from that age for the balance of life. Column 3, the monthly rates as derived from the annual rates with allowance for slight loss due to that method of payment. These two columns are the basis of calculating columns 4 and 5. Column 4 shows a modification of the natural step-rate by means of an accumulation of 15 cents per month which is used to reduce the level cost from age 61 to $3.00 per month. Column 5 a similar modification but with an accumulation of 30 cents per month and a level cost from age 61 of $2.50 per month. Under either of these plans all members pay the same rates at the same attained ages. The purpose in view in these tables is to have a plan that requires but little detail in its operation so as to be readily comprehended by the officers of the local lodges.

The step-rate plan, as shown in Column 2, Table III, can be modified to meet the necessities of different societies by varying the amount of accumulation. The following table is submitted as a basic table for that purpose. It shows how an accumulation of $1.00 per annum, paid in monthly installments, may be used to reduce the level cost after age 61, from the level rate of $54.01. The table shows the amount of such reduction; based on age at entry, giving to each member the full benefit of the term of membership. Thus the member entering at 21 would secure an annual reduction of $11.61, giving an annual cost from age 61 of $42.40. The member entering at 36 would secure a reduction of $4.71 giving annual cost from age 61 of $49.30. The adjustment of annual cost after age 61, would only have to be made when the members reach age 61, the rates being the same for same attained ages from age 21 to 60.

With this table as a basis, the annual accumulation necessary to secure greater reduction can be calculated. If the accumulation was $2.00 per annum, the reduction would be twice that of the table, and so for any other amount of accumulation.

At age 61, the level rate is $54.01. At each age at entry $1.00 additional to natural step-rate paid as a special accumulation, will give the following annual reduction from age 61:

The rates recommended by the Committee on Assessments, in the suppressed report to which reference has been made, were as follows.