Life insurance; legal net values; a popular treatise on the method of computing the net value of life policies and showing the nature and proper uses of this fund

Preface

The following series of Articles, recently published in ‘The Spectator,” are offered to the public in pamphlet form, with the hope .that they will be found useful to those who do not already clearly understand the simple elementary principles upon which life insurance calculations are made— after a table of mortality and rate of interest have been determined upon as the bases for the computations. These principles apply to any table of mortality and rate of interest. Without waiting the completion of the ideal table, for which the actuaries seem to be still zealously searching, by means of complex mathematics applied to statistics obtained from the experience of companies, we have endeavored to explain the method of using the table of mortality now designated by law believing it to be sufficiently accurate for present practical purposes.

Contents

I. The Method of Computing the Legal Net Value of a Life Policy, assuming that the Net Annual Premiums and the Net Value of a life series of Annual Payments of $i are known

II. The disposition that ought to be made of the Legal Net Value in case a Renewal Premium is not paid when due. Interest. Mortality Tables. The method of calculating the net present value of a life series of Annual Payments of $i each

III. Net Cost, at any age, of Insurance for one year. Net single premium

I

HE law requires that a life insurance company shall hold l — invested in certain prescribed classes of securities — the “ net value” of every policy it has in force. A popular explanation of the simple principles upon which calculations of these values are based, will enable those interested to form an intelligent opinion in regard to the disposition that ought to be made of this fund in case the policy to which it pertains is not continued in force by the payment of a premium when due. The aggregate accrued net value of life insurance policies now in force in this country is more than three hundred million dollars, and the amount is rapidly increasing. The policyholders furnish the money upon which this gigantic business is conducted, and every policyholder ought to have some idea of the practical meaning of this law, which was enacted for the government of the corporation that has the handling and control of the money he pays to secure his heirs from poverty and want.

For present purposes of illustration it is not now necessary to explain in detail the simple arithmetical principles used in calculating net premiums in life insurance. The net annual premium that will — if paid at the time the policy issues and payment is renewed at the beginning of each succeeding policy-year that the policyholder is alive — be just sufficient to insure a given amount to his heirs at the end of any year in which he may die, is readily susceptible of definite arithmetical computation, based upon any table of mortality and rate of interest.

Assuming that the actuaries’ table of mortality and four per cent interest are designated by law as the bases for calculating the “ net value ” of a life insurance policy, the net annual premium, at age 20, to insure $1,000 for life is $12.94. The net annual premium takes no account of expenses, profits, or contingencies — these are provided for by an addition called “ loading.”

The policyholder, at age 20, pays the net annual premium $12.94, and is insured for one year. If he is alive at age 21 and pays at that time a renewal net annual premium, $12.94, he is insured the second year. But the net annual premium necessary at age 21 to insure $1,000 for life is $13.27. The insurer who accepts at age 21 a net annual premium $12.94 to insure $1,000 for life, must have on hand, to the credit of this policy, an amount equivalent to the value at this age of a life series of annual payments each equal to $13.27 less $12.94 = $0.33. The value at age 21 of a life series of annual payments each equal to $0.33 is $6.37. This $6.37 is the u net value ” which the law requires the company to hold for this policy at age 21. The net annual premium which was paid at age 20 was sufficient, on the legal data, to pay net cost of insurance on this policy the first year and leave $6.37 in the hands of the company to the credit of this policy at the end of the year.

If this policyholder is alive at age 22, and pays at that time a renewal net annual premium $12.94, the company can insure him for this net annual premium, notwithstanding the fact that the net annual premium necessary at this age to insure $1,000 for life is $13.61. This can be done, however, only in case the company holds at this time to the credit of this policy an amount equal to the value at age 22 of a life series of annual payments each equal to $13.61 less $12.94 = $0.67. The value at age 22 of a life series of annual -payments each equal to $0.67 is $12.96. This $12.96 is the legal net value of this policy at the end of the second year. It was formed partly by the legal net value $6.37 held by the company to the credit of this policy at the end of the first year, and partly by a portion of the net annual premium paid at the beginning of the second year. The other portion of the second net annual premium was just sufficient to pay net cost of insuring this policy during the year between age 21 and 22.

In like manner at the beginning of each policy-year that this policyholder is alive the net annual premium, $12.94, will effect his insurance, provided the company holds the prescribed legal net value ; because this net annual premium and this net value are together equivalent at each age to the net annual premium necessary at each age to insure $1,000 for life.

In case a policyholder dies in any year, the net value of his policy is used by the company in part payment of his death loss ; but if the policyholder is alive, and continues his policy in force by the payment of a renewal premium at the beginning of the next policy-year, the net value is held by the company, and is to be used in part payment of the policy to which it pertains when this policy matures. The net value increases every year that a policy continues in force ; at the table limit of mortality it is equal to the amount of the policy. As the net value increases, the amount the company has at risk on this policy diminishes every year that the policy continues in force, and becomes zero at the end of the table of mortality. The net annual premium is composed of two distinct parts, one of which goes to pay net cost of insuring the amount at risk each year, the other goes to form “ net value.” After a policy has been in force for a number of years the net cost of insuring the amount at risk becomes greater than the net annual premium — in this case the balance of the net cost is made good from interest on the net value during the year — leaving, however, enough of this interest to bring the net value at the end of the year up to the amount required by law.

Returning to the policy for $1,000 taken out at age 20 ; suppose this policyholder is alive at age 60 and has paid his annual premiums regularly every year. The net annual premium necessary at age 60 to insure $1,000 for life is $57.55 ; but the company can, at this age, insure this policyholder for the net annual premium, $12.94, because the company is required to hold to the credit of this policy an amount equal to the value at age 60 of a life series of annual payments, each equal to $57.55 less $12.94 = $44-6 1. The value at age 60 of a life series of annual payments each equal to $44.61 is $464.39. This is the “ net value ” of this policy at age 60. This policyholder has paid year by year, in advance, the net cost of insuring the amount at risk on this policy each year, and he has paid every year — out of the “ loading” part of his premiums — his proportion of the expenses, profits, and contingencies. In addition, the company holds $464.39 to the credit of this policy, which amount enables the company to continue the insurance upon receiving the net annual premium, $12.94, although the net annual premium necessary at this age to effect the insurance is

$57-55-

Now suppose this policyholder does not pay his renewal annual premium at age 60, what ought to be done with the $464.39? Many companies settle this question by so wording the policy that the insured agrees to forfeit all his interest in it in case he fails at any time to pay a renewal premium when due. This forfeiture is indefensible in equity, and not called for by any proper consideration for the rights of those policyholders who continue to pay renewal premiums.

The net value of a life insurance policy at the end of any policy year is as clearly a part of the policyholder’s payments for his policy as the renewal premium then due will be if he pays it. The net value was formed out of portions of the net annual premiums he has already paid, and for which he has received no equivalent. The legal net value may, therefore, be considered “ unearned premium.”

If there is no legal contract to the contrary, this unearned premium — which the companies call “ reserve,” but which the law calls “net value” — should be applied to the purchase of paid-up insurance for the policyholder who does not continue his original policy in force by the payment of a renewal premium when due.

Life insurance corporations should not be permitted, in future contracts, to appropriate to themselves the accrued legal net value ; but the law should require that paid-up insurance be issued for such an amount as the net value — less any indebtedness of the policyholder to the company — will purchase at the time a renewal premium is due and not paid. In applying the sum thus determined to the purchase of paid-up insurance, fair allowance should be made for expenses and contingencies connected with the new paid-up policy.

II

THOROUGH comprehension of the method of computing these net values requires a knowledge of the arithmetical principles used in calculating the present value, at any age, of a life’s series of annual payments of $i each, and calculating the net premium that will, at any age, insure a given amount. Before explaining these principles allusion will be made to a decision of the Supreme Court of the United States, which bears upon the question of the disposition that ought to be made of the accrued legal net value in case a renewal premium is not paid when due.

There were several cases before the court in which Southern policyholders had during the war, failed to pay renewal premiums, and had died before the end of the war. In reference to the legality of this forfeiture clause the court held that :

“ Time is material and of the essence of the contract, and non-payment at the day involves absolute forfeiture, if such be the terms of the contract, as is the case here. Courts cannot with safety vary the stipulation of the parties by introducing equities for the relief of the insured against their own negligence.”

This opinion settles the question of the legality of this forfeiture clause in life insurance policies, in cases where such forfeiture is not prohibited by the charter of the company, or by the laws of the State under which the company was incorporated, or of the State in which the insurance contract was made. The court further held —

“ That such failure being caused by a public war, without the fault of the assured, they are entitled, ex cequo et bono , to recover the equitable value of the policies, with interest from the close of the war.” * * “In each case the rates of mortality

and interest used in the tables of the company will form the basis of the calculation.”

The court illustrated its opinion by an example in which the policy is supposed to have been issued at age twenty-five, for the annual premium due at that age, and the policyholder had reached the age forty-five when he failed to pay his renewal premium because of a public war. On this point the court says :

“ The present value of the amount assured is exactly represented by the annuity which would have to be paid on a new policy, or $38 per annum in the case supposed, when the party is 45 years old, while the present value of the premiums yet to be paid on a policy taken by the same person at 25 is but little more than half that amount. To forfeit this excess, which fairly belongs to the assured, and is fairly due from the company, and which the latter actually has in its coffers, and to do this for a cause beyond individual control, would be rank injustice. It would be taking away from the assured that which had already become substantially his property. It would be contrary to the maxim, that no one should be made rich by making another poor.”

The court decided : —

First. u The policies in question must be regarded as extinguished by the non-payment of the premiums, though caused by the existence of the war, and that an action will not lie for the amount insured thereon.”

Second. “ That the assured 4 are entitled to recover the equitable value of the policies with interest from the close of the war.’ ”

In other words, the non-payment of the renewal premium having been caused by war, this forfeiture clause became inoperative, and the case was decided as if this condition was not in the contract. The forfeiture of the accrued legal net value has no foundation in equity, and holds good against the assured only because the contract calls for it. The original policy ought not to continue in force if a renewal premium is not paid when due ; but these corporations, created by State authority, for the purpose of executing certain important trusts for widows and orphans, should not be permitted, in future contracts, to appropriate to themselves the accrued legal net value of any of their policies.

In calculating the net value of a policy we need to know the net annual premium that will, on the legal data, effect the insurance at the age of the policyholder at the time for which the policy is being valued, and to subtract from this the net annual premium due to the age of the policyholder at the time his policy was issued. This gives the difference between the net annual premium required at the time for which the policy is being valued and the net annual premium the insured will pay. The question then arises : What is the present value, at the time for which this policy is being valued, of a life series of annual payments each equal to this difference ?

Interest . — In making these computations, the following arithmetical rule for determining the present value of $i, payable at the end of any given number of years, finds constant application. “ The amount that will, at any named rate of interest per annum, become $i in one year is obtained by dividing $i by unity plus the rate of interest. The amount that will, at this rate of interest, compounded annually, become $i in any designated number of years is obtained by raising the amount that will become $i in one year to a power the exponent of which is the number of years.” In illustration of this rule, suppose the rate of interest is 4 per cent per annnm. The amount that will, when increased by interest at this rate for one year, become $ 1 is equal to The amount that will, at the same rate of interest per annum, compounded annually, become $1 in two years is equal to il x ^ And so on’ multiplying by r-h for each additional year that interest is to be compounded. These calculations have been made, and the results are placed in tables which show the amount that will, if invested at four per cent per annum compounded annually, become one dollar in any designated number of years, from one to one hundred.

Mortality Tables. — The amount that will, at the named rate of interest, become one dollar in a designated number of years being known, the second step in these calculations will have been made when we determine the number of dollars that will be required at the end of the designated number of years. Mortality tables enable us to determine this number of dollars. The actuaries’ table of mortality was deduced from many years’ experience of seventeen principal life insurance companies of Great Britain. This table shows that out of one hundred thousand persons living at age ten, the number of these that will die between age ten and eleven is 676. This leaves 99,324 living at age eleven ; out of which number 674 will die between age eleven and age twelve. And so on, the table shows the number that will be living at each age, and the number of these that will die before an age one year greater.

A Life Series of Annual Payments of One Dollar Each. — The Actuaries’ table of mortality and four per cent interest being designated as the bases upon which calculations of net values in life insurance shall be made, the net present value, at any age, of a life series of annual payments of one dollar each is found as follows : Suppose the age is 40, and that each of the 78,653 persons living at this age, as shown by the tables of mortality, are to receive one dollar annually for life, the first payment being immediate. The insurer will require $78,653 in hand in order to enable him to make the first payment. The table shows that of those living at age 40 there will be 77,838 alive at age 41. Therefore the insurer will require, at the beginning of the second year, $77,838 in order to pay at that time one dollar to each of those that will then be alive. The amount that will, at four per cent, become one dollar in one year is expressed by AL. Therefore, 77,838 is the net amount the insurer ought to receive at the time the contract is entered into, to enable him to pay, at the beginning of the second year, one dollar to each of those that will then be alive.

In like manner we find the net amount the insurer ought to receive at the time the contract is entered into, in order that he may have at the beginning of each succeeding year $1 for each of those that will then be alive. The sum of these yearly values to the table limit of age is the net amount that will, if paid in hand at age 40, enable the insurer to pay $1 annually for life to each of the 78,653 persons living at age 40. The seventy-eight thousand six hundred and fifty-third part of this amount is what each one ought to pay. These calculations have been made, and they show that the net present value at age 40 of a life series of annual payments of $1 each is $16.09. Similar calculations have been made at each age, and the results are placed in tables.

III

NET Cost of Insurance for One Tear . — Suppose the age of the insured is 40. The net amount that will insure $1000 to be paid to his heirs at the end of the year, in case he dies between age 40 and 41, is computed as follows : The table of mortality shows that out of 78,653 persons living at age 40 the number that will die between age 40 and 41 is 815. In case the whole 78,653 living at age 40 are insured for $1000 each for one year, the insurer will require $815,000 at the end of the year to enable him to pay $1000 at that time to the heirs of each of those that die in that year. The amount that will, at four percent, become $1 in one year is expressed by • therefore, the amount the insurer ought to receive at the beginning of the year is expressed by 2L x 815,000. Divide this by the whole number living at age 40 and we have the net amount each ought to pay. Performing the operations indicated we find this amount is $9.96.

In a similar manner the calculations have been made at the other ages, and the results are placed in tables which show the net amount that will at each age insure $1000 for one year.

Net Single Premium, that will Insure $1000 for Life. — Again assume that the age of the insured at the time the policy is issued is 40. We have already found the net amount that will, if paid at age 40, insure $1000 for one year. To find the net amount that will at age 40 insure $1000 to be paid to the heirs of the insured at the end of the second year in case he dies between age 41 and age 42 ; obtain from the table of mortality the number of deaths between age 41 and age 42 — this number is 826. Therefore, the insurer will require $826,000 at the end of two years, in order to pay at that time $1000 to the heirs of each of those that will die between age 41 and age 42. The amount that will at four per cent per annum, compounded annually, become $1 in two years is expressed by -I- Therefore the amount the insurer ought to receive, at the time the contract is entered into, to enable him to pay at the end of two years $1000 to the heirs of each of those that will die between age 41 and age 42 is expressed by JL _L x 826,- 000. Divide this by the whole number living at age 40 and we have the net amount each ought to pay at age 40 in order to insure $1000 to his heirs in case of his death between age 41 and age 42.

In a similar manner we compute the net amount that will at age 40 insure $1000 in case the insured dies between age 42 and age 43, and so on for each year to the table limit. It will be noticed that the table of mortality is arranged by years — interest is at a certain rate per cent per annum compounded annually — the calculations are made for insurance in each separate year without reference to insurance in any other year, and these separate yearly insurances may or may not be combined, depending upon the agreement between the insurer and the insured. When these separate yearly insurances for every year to the table limit of age are added together their sum is called the net single premium at age forty for whole life insurance of $1000. These calculations have been made and the result shows that at age 40 the net single premium that will insure $1000 for life is $381.04.

In a similar manner these calculations have been made at each age, and the results are shown in tables.

The Net Annual Premium. — In illustration, again assume that the age is 40. We have just seen that at this age the net single premium that will insure $1000 for life, is $381.04 ; and we have previously found that the net value at age 40 of a life series of annual payments of $1 each is $16.09. These data furnish the means for forming a proportion, the fourth term of which, when the proportion is solved, will show the amount of the net annual premium. The proportion is as follows :

$16.09 to $3^I-°4 as $1 is to the fourth term.

Solving this proportion, we find the fourth term is $23.67. This is the net annual premium that will, at age 40, insure $1000 for life. This is so, because a life series of annual payments of $1 each, beginning at age 40, is equivalent to $16.09 in hand at that age ; and the proportion shows that this being true, $23.67 paid annually for life, beginning at age 40, is the equivalent of $381.04 in hand at that age. But $381.04 paid at age 40 will insure $1000 for life; therefore, its equivalent in annual payments of $23.67 each will insure $1000 for life.

In a similar manner the calculations have been made at each age, and the results are shown in tables.

Net Cost of Insuring the Amount at Risk. — The legal net value of an ordinary life policy, at the end of any policy year, is the value at that time, of a life series of annual payments each equal to the difference between the net annual premium due to the age at which the policy is being valued, and the net annual premium the insured pays. The amount the company has at risk any year is the amount of the policy less the legal net value at the end of that year. Having calculated the legal net value at the end of any policy year the amount the company has at risk during that year becomes known. It is now proposed to illustrate the method by which we determine what part of each net annual premium goes to pay net cost of insurance, and what part goes to form legal net value.

In illustration : assume that an ordinary whole life policy for $1000 is taken out at age 42, the net annual premium at this age (actuaries’ table of mortality and 4 per cent interest) is $25,554. (See table.) The legal net value of this policy at the end of the first year, computed by the method just referred to, is $158.51. The amount at risk on this policy during the year between age 42 and 43 is $1000 less $158.51 = $984,149. The table shows that at age 42, the net amount that will insure $1000 for one year is $10,476.

From this we find that at age 42, the net amount that will insure $984,149 for one year is $10.31. Subtract this from the net annual premium $25,554, Pa*d at age 42, and we have $15,244 of this premium left after providing net cost of insuring the amount at risk during the year. This $15,244, increased at 4 per cent will become $15.85 at the end of the year. This $15.85 is the legal net value at the end of the year, as calculated by the method above referred to.

In like manner that part of the net annual premium at age 43 which pays net cost of insurance on the amount at risk during the year between age 43 and age 44 may be determined. When we have found this amount its value at age 42 is obtained by multiplying it by Bear in mind that the premium at age 43 is to be paid only in case the insured is then alive. Therefore, the above amount must be multiplied by the number living, as shown by the table at age 43, and this product divided by the number living at age 42, in order to determine the net present value* — at the time this policy is issued — of that part of the net annual premium at age 43, which will pay net cost of insuring the amount at risk on this policy during the year between age 43 and age 44. These calculations are made in a similar manner for each year to the table limit of age.

The net present value, at age 42, of u the normal contributions this policyholder is liable to have to make in payment of death claims other than his own” is obtained by finding the sum of the foregoing amounts for each year to the table limit. This sum will, if paid at age 42, insure the amount at risk on this policy every year to the table limit. That part of the $1000 insurance not included each year in the amount at risk is provided for by the legal net value at the end of that year.

In a manner similar to that indicated above, for age 42, the net value of these “ contributions ” may be calculated for any

age-

It is important that every policyholder, in a life insurance company, should know that the contribution he is liable to have to make in payment of death claims, other than his own, is balanced by the liability of the other policyholders to contribute to his death claim. These other policyholders are not entitled to the contributions he is liable to make in payment of their death claims unless their liability to have to contribute to his death claim is clearly acknowledged. This principle will be again referred to in another connection.