LIFE INSURANCE LEGAL NET VALUES A Popular Treatise METHOD OF COMPUTING THE NET VALUE OF LIFE POLICIES AND SHOWING THE NATURE AND PROPER USES OF THIS FUND. By GUSTAVUS W. SMITH, Formerly Insurance Commissioner of Kentucky. NEW YORK: PUBLISHED BY THE SPECTATOR COMPANY, No. 16 Dey Street. 1879. JK I ' Digitized by the Internet Archive in 2017 with funding from University of Illinois Urbana-Champaign Alternates https://archive.org/details/lifeinsurancelegOOsmit 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 calcu- lations 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 ex- perience 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 for whole life insurance. Net Annnal Premium for whole Life In- surance. Net cost each year of insuring the amount at risk IV. Tables I. to VI. inclusive. The method of determining that part of the net single premium that will go each year to pay net cost of insuring the amount at risk that year V. The method of computing the Legal Net Value in case the company has contracted to furnish insurance for net premiums, less than those called for by the data prescribed by law. Comments upon net valuations . . . VI. The method of determining, at any age, the net present vaiue of $i to be paid in any designated number of years, provided the person to whom the payment is to be made is alive at the end of that time. Masseres, 1783. LIFE INSURANCE LEGAL NET VALUES . 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 ex- planation 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 in- terest. 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 pre- mium, at age 20, to insure $1,000 for life is $12.94. The net 6 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 7 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 in- creases, the amount the company has at risk on this policy diminishes every year that the policy continues in force, and be- comes 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 inter- est 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 ; sup- pose 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 8 amount at risk on this policy each year, and he has paid every year — out of the “ loading” part of his premiums — his propor- tion 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 re- ceived 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 pre- mium 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 in- debtedness of the policyholder to the company — will purchase at the time a renewal premium is due and not paid. In ap- plying the sum thus determined to the purchase of paid-up in- surance, fair allowance should be made for expenses and con- tingencies connected with the new paid-up policy. 9 II. T HOROUGH 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 ac- crued legal net value in case a renewal premium is not paid when due. There were several cases before the court in which South- ern policyholders had during the war, failed to pay renewal premiums, and had died before the end of the war. In refer- ence 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 negli- gence.” This opinion settles the question of the legality of this for- feiture clause in life insurance policies, in cases where such for- feiture 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 re- cover 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, 10 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 repre- sented 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 com- pany, and which the latter actually has in its coffers, and to do this for a cause beyond individual control, would be rank in- justice. 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 ex- tinguished 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 inop- erative, 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 11 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, pay- able at the end of any given number of years, finds constant application. “ The amount that will, at any named rate of in- terest 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. Mor- tality 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 com- panies of Great Britain. This table shows that out of one hun- 12 dred 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 num- ber 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 be- ing 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 sec- ond 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. 13 III. N ET 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 1.04 1.04* O 7 14 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 com- pounded 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 pay- ments of $1 each, beginning at age 40, is equivalent to $16.09 in hand at that age ; and the proportion shows that this being 15 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 equiva- lent 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 com- pany 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 in- sure $984,149 for one year is $10.31. Subtract this from the net annual premium $25,554, P a *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 calcu- lated by the method above referred to. In like manner that part of the net annual premium at age 43 16 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 num- ber 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 contribu- tions this policyholder is liable to have to make in payment of death claims other than his own” is obtained by find- ing 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 con- tribute 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. * See Article VI. 17 IV. T HE method by which we compute the amount that will, when increased by interest at any designated rate per annum, compounded annually, become one dollar in a given number of years has already been stated. INTEREST. Table I — Shows the amount that will, when increased at 4 per cent per annum, compounded annually, become $1 in any designated number of years, from one to one hundred, in- clusive : J Years. Amount at \per cent. Years. Amount at 4 per cent. Years. Amount at 4 per cent. Years. Amount at 4 per cent. 1 $0.9615385 1 26 $0.3606892 5 i $0 1353006 76 $0.0507535 2 0.9245562 27 0.3468166 52 0.1300967 77 0.0488015 3 0.8889964 28 0.3334775 i 53 0.1250930 78 0.0469245 4 0 8548042 j 29 0.3206514 i 54 0.1202817 79 0.0451197 5 0.8219271 30 0.3083187 55 0.1156555 80 " 0.0438843 6 0.7903145 3 i 0 2964603 5 6 0. 1 1 12072 81 0.0417157 7 0.7599178 32 0.2850579 . 57 0. 1069300 82 0.0401 1 12 8 0.7306902 33 0.2749242 58 0.1028173 83 0.0385685 9 0.7025867 34 0 2635521 59 0.0988628 84 0 0370851 10 0.6755642 35 0.2534155 ! 60 0.0950604 85 0.0356587 11 0.6495809 36 0.2436687 61 0 0914042 86 0.0342873 12 0.6245970 37 0.2342968 62 0 0878887 87 0.0329685 13 0 6005741 38 0.2252854 63 0.0845083 88 0.0317005 14 0.5774751 39 0.2166206 64 0.0812580 89 0.0304812 15 0.5552645 40 0.2082890 65 0.0781327 90 0.0293080 16 0.5339082 4 i 0.2002779 66 0.0751276 9 i 0.0281816 1 7 0.5133732 42 0.1925749 67 0.0722381 92 0 0270977 18 0.4936281 43 0.1851682 68 0.0694597 93 0.0260555 19 0.4746424 44 0 1780463 69 0.0667882 94 0.0250534 20 0.4563869 45 0.1711984 70 0.0642194 95 0.0240898 21 0.4388336 46 0 1646130 7 i 0.0617494 96 0.0231632 22 0.4219554 47 0.1582826 72 0.0593744 97 0.0222723 23 0.4057263 48 0.1521948 73 0.0570908 98 0.0214357 24 0 3901215 49 0.1463411 74 0.0548950 99 0.0205920 25 0 3751168 50 0.1407126 75 0.0527837 100 0.0198000 18 ACTUARIES* TABLE OF MORTALITY. Table II — This table was compiled from observation and experience, as previously explained. It shows that out of ioo.ooo insured persons living at age io, the number of these that will die between age io and age n is 676 ; the number that will die between age 11 and age 12 is 674 ; and so on giv- ing the number that will die each year to the table limit of age. The table also shows the number that will be living at each age. Age. Living. Deaths. Age. Living. Deaths. Age. Living. Deaths. 10 100,000 676 4 1 78,653 815 70 35,837 2.327 11 99.324 674 4 i 77.838 826 7 i 33 , 5 io 2,351 12 98.65 - 672 42 77 012 839 72 3 i 159 2 362 13 97978 671 43 76,173 857 73 28,797 2,358 14 97.307 671 44 75.316 881 74 26,439 2,339 IS 96.636 671 45 74.435 909 75 24,100 2,303 16 95.965 672 46 73,526 944 76 21,797 2,249 1 7 95.293 673 47 72.582 981 77 19,548 2,179 18 94,620 675 48 71,601 1,021 78 17.369 2,092 19 93 94 - 677 49 70 580 1,063 79 15.277 1,987 20 93,268 680 50 69.517 1,108 80 13,290 1,866 21 92,588 683 5 i 68,409 1,156 81 11,424 1,730 22 91.905 686 52 67,253 1,207 82 9.694 1,582 23 91,219 690 53 66,046 1,261 83 8,112 1,427 24 90.529 694 54 64,785 1,316 84 6,685 1,268 25 89,835 698 55 63,469 1.375 85 5 , 4 i 7 1, hi 26 89,137 703 56 62,094 1,436 86 4-306 958 27 88,434 708 57 60,658 1,497 87 3,348 811 28 87,726 714 58 59 .i 6 i 1,561 88 2 537 673 29 87,012 720 59 57 , 6-0 1,627 89 1 864 545 30 86,292 727 60 55 973 1,698 90 1.319 427 3 i 85.565 734 61 54 275 1,770 9 1 892 322 32 84.831 742 62 52 505 1,844 92 ! . 570 231 33 84,089 750 63 50,661 i, 9 I 7 93 339 155 34 83.339 758 64 48,744 1,990 94 1 184 95 35 82,581 767 65 46,754 2 061 95 89 52 36 81,814 776 66 44.693 2,128 96 37 24 37 81,038 785 67 42,565 2,191 97 13 9 38 80,253 795 68 40.374 2,246 98 4 3 39 79,458 805 69 38,128 2,291 99 1 1 The methods used in computing the values shown in the following tables have already been explained and illus- trated : 19 Table III — Shows the net value at each age of a life series of annual payments of $i each, the first immediate — Actuaries* Table of Mortality — interest at four per cent. Years . Amount at \per cent . Years . Amount at 4 per cent . Years . II Amount at 4 per cent . Years . Amount at \per cent . 10 $20.4536 33 $17.5196 56 $11.6698 79 $4 . 8986 11 20.3694 34 17.335° 57 H -3593 80 4.6607 12 20.2818 35 17.1443 58 11.0463 81 4.4290 13 20 . 1907 36 16.9476 59 10. 7311 82 4.2026 14 20.0959 37 16 . 7443 60 10.4147 8 J 3.9802 15 19.9976 38 16 . 5342 61 10.0977 84 3.7611 16 I 9 - 89 S 7 39 16.3172 62 9 - 78 o 5 85 3-5436 17 19.7901 40 16 . 0929 63 9.4641 86 3-3279 18 19 . 6807 4 i 15 . 8610 64 9.1489 87 3-1138 19 I 9-5675 42 15.6212 65 8 8356 88 2 . 9012 20 19.4504 43 15 . 3736 66 8.5248 89 2.6911 21 I 9-3293 44 15.1186 67 8 . 2170 90 24854 22 19 . 2042 45 14.8571 68 7.9130 9 i 2.284? 23 19.0747 46 14.5896 69 7.6130 92 2.0902 24 18.9410 47 14.. 3170 70 7.3172 93 1 . 9065 25 18.8027 48 14 .. 0304 7 i 7 . 0261 94 1.7369 26 18.6598 49 13 . 7572 72 6.7400 95 1 5843 27 18.5122 50 13 4/703 73 6-4593 96 1.4618 28 18.3597 5 i * TV 13 . 1792 74 6. 1840 97 1.3670 29 18 2022 52 12.8841 75 5 - 9 I 46 98 1 . 2404 30 18.0397 53 12. ^ 8^3 76 5-6512 99 1. 0000 3 i 17.8718 54 • J W JJ 12 . 2832 77 5-3938 32 17-6985 55 11.9779 78 5.1428 Table IV — Shows the net cost of insuring $1000 for one year, at different ages, from 20 to 70, inclusive. (Actuaries* Table of Mortality — interest at four per cent.) Age . Age . J Age . 20 $7,010 37 $9,314 54 $i 9-532 21 7093 38 9 525 55 20.831 22 7.177 39 9.741 56 22.237 23 7-273 40 9963 57 23.730 24 7-371 4 i 10.204 58 25 371 25 7-471 42 10.476 59 27.160 26 7-583 43 10.818 60 29.169 27 7.698 44 11.247 61 31-357 28 7.826 45 11.742 62 33-770 29 7-956 46 12.345 63 36.384 30 8.101 47 12.996 64 39-255 3 i 8.248 48 13.711 65 42.386 32- 8.410 49 14.482 66 45.782 33 8.576 5 o 15-326 67 49.494 34 8.746 5 i 16.248 68 53-490 35 8931 52 17257 69 57776 36 9.122 53 18,359 70 62.436 20 Table V— Shows the net single premium that will, at differ- ent ages, from 20 to 70, inclusive, insure $1000 for life. (Actu- aries’ Table of Mortality — interest at four per cent.) Age . Age . Age . 20 $251,907 37 $ 355-989 54 $ 527,567 . 21 256.564 38 364.065 55 539-312 22 261.377 39 372.414 56 55 I-I 57 23 266 357 40 381.040 57 563-103 24 271 500 4 i 389.960 58 575-142 25 276.8x6 42 399 -I 83 59 587257 26 282.312 43 408.709 60 599-433 2 7 287,990 44 418.515 61 611.628 28 293.856 45 428.571 62 623 826 29 299.913 46 438.862 63 635-995 3 ° 306.168 47 449 346 64 648 120 3 1 312 624 48 460.022 65 660.171 32 319.289 59 470.878 66 672.124 33 326.167 50 481.906 67 683.968 34 333 - 26 7 5 i 493.107 68 695-654 35 340.600 52 504.460 | 69 707 192 36 348 170 53 5 I 5-949 70 718.569 Table VI — Shows the net annual premium that will, at different ages, from 20 to 70, inclusive, insure $1000 for life. (Actuaries’ Table of Mortality— interest at four per cent.) Age . Age . Age . 20 $12,948 37 $21 260 54 $42,950 21 I 3-273 38 22 018 55 45-025 22 13 610 39 22 823 56 47 230 23 24 13- 963 14 - 334 40 4 1 23.677 24.586 57 58 49-571 1 52 067 25 14.722 42 25 554 59 54 724 26 15.129 43 26.585 60 57-556 27 28 15-557 16 005 44 45 27.682 28 845 61 62 60.572 63.782 29 16 477 46 30.080 63 67.199 3 ° 16.972 47 3 I -385 64 70.841 31 17.492 48 32 767 65 74.718 32 18.040 49 34 227 66 78.846 33 18.616 50 35 775 67 83-237 34 19 225 5 i 37-415 68 87 - 9 t 3 35 19 866 52 39 .I 5 I 69 92 892 36 20544 53 40.996 70 98 202 21 The tables given above present an apparently formidable array of figures, but on examination it will be found that all the tables are arranged by successive whole years, and that opposite each age or number of years will be found in its pro- per column certain specific information. In illustration take age 30. Table I shows that $0.3083187 will, at 4 per cent, compounded annually, become $1 in 30 years. Table II shows that at age 30 there are 86,292 per- sons living out of 100,000 that were living at age 10 ; and that out of this number living at age 30 the number that will die before age 31 is 727. It also shows that out of the 100,000 living at age 10 the number of these that will die between age 30 and age 31 is 727. Table III shows that at age 30 the net present value of a life series of annual payments of $1 each is $18.0397. Table IV shows that at age 30 the net cost of insuring $1000 for one year is $8,101. Table V shows that at age 30 the net single premium that will insure $1000 for life is $306,168. This is for paid up insurance. Table VI shows that at age 30 the net annual premium that will insure $1000 for life is $16,972. In like manner, we can readily find in the tables information similar to the above, by looking in the proper column, opposite to the designated number of years or age of the insured. * In illustration : Suppose that it is required to find the legal net value at age 60 of a whole life policy for $1000 issued at age 20. The net annual premium necessary at age 60 to insure $1000 for life is $57*556 (see table VI). But the policyholder who took out his policy at age 20 pays a net annual premium of $12,948 only (see same table). The difference between the net annual premium necessary at age 60 to effect this insurance, and the net annual premium the insured pays at this age, is $57,556 less $12,948 = $44,608. The value at age 60 of a life series of net annual premiums each equal to $44,608 is obtained by using table III, from which we find the value at age 60 of a life series of annual payments of $1 each is $10.4147. Multiply this by $44,608 and we have $464,578, which is the net value at age 60 of a life series of annual pay 22 ments, each equal to the difference between the net annual premium necessary at age 60 to insure $1000 for life, and the net annual premium the insured will pay. This $464,578 is the legal net value of this policy at age 60. And this is the amount the law requires the company to hold to the credit of this policy at this age. If by the terms of the contract the insured was not required to pay any more net annual premiums — in other words, if this policy was full paid at age 60 — the legal net value would then be the net present value at age 60 of a life series of annual pay- ments, each equal to $57,556. The net present value at age 60 of a life series of annual payments of $1 each is $10.4147 (see table III). Multiply this by $57,556, and we have $599.43, which is the net single premium at age 60. Net cost each year of insuring the amount at risk . — Sup- pose a paid-up whole life policy for $1000 is issued at age 20. The net single premium in this case is $25 1 .907 (see table V), and this is the net amount that will, if paid at age 20, insure $1000 to the heirs of the insured at the end of any year in which he may die. Table V shows that the net single premium at age 21 necessary to insure $1000 for life is $256,564. The net single premium ($251,907) paid at age 20, will pay net cost of insuring the amount at risk on this policy during the year between age 20 and age 21, and leave in the hands of the company at the end of the year the net single premium $256,564, which is the net amount requisite at this age to insure $1000 for life. Since this is a paid-up policy its legal net value af^age 21 is the net single premium that will at that age insure $1000 for life. If the policy continues in force the company must hold at the end of the year the net single premium, $256,564, after net cost of insuring the amount at risk during the year has been provided for out of the net single premium paid at the begin- ing of the year. The amount at risk on this policy during the year between age 20 and age 21, is equal to the amount of the policy less the legal net value at the end of the year. It is, therefore, $1000 less $256,564 = $743,346. The net amount that will at age 20 insure $1000 for one year is $7 01 (see table IV). From this we find that the net amount that will, at age 23 20, insure $743,346 for one year is $5,211. This is the amount necessary at age 20 to insure the amount at risk on this policy during the year between age 20 and age 21. Subtract this from the net single premium paid at age 20, and we have $251,907 less $5,211 = $246,696, which is that part of the net single premium paid at age 20 which goes to form legal net value for this policy at the end of the year. Increase $246,696 by 4 per cent, and we have $256,564 at the end of the year. This is the net single premium at age 21 (see table V). Each policyholder living at age 20 contributes $5,211 out of the net single premium, $251,907, to pay death claims during the year. Each of those who die between age 20 and age 21 contributes to his own death claim in addition to this $5,211, the legal net value of his policy at the end of the year. The contribution, $5,211, made at the beginningof the year by each policyholder living at that time, insures the amount at risk on his policy during the year ; in other words, pays net cost of insuring $743-346 during the year between age 20 and age 21. The legal net value at the end of the year makes up the $1000 death claim. If the policyholder is alive at age 21 the company holds $256,564 to the credit of his policy, after deducting net cost of insuring the policy for the year between age 20 and age 21. This $256,564 must provide net cost of insuring this policy during the year between age 21 and age 22, and leave in the hands of the company the legal net value at the end of the year. This legal net value is the net single premium that will at age 22 insure $1000 at the end of any year in which the insured may die. Table V shows that this amount is $261,377. The amount at risk during the year between age 21 and age 22 is $1000, less $261,377 = $738,623. Table IV shows that at age 21 the net cost of insuring $1000 for one year is $7,093. From this it follows that at age 21 the net cost of insuring $738,623 for one year is $5,239. Deduct this from the net single premium at age 21, and we have $256,564, less $5,239 = $ 2 5 I -3 2 5’ which is that part of the net single premium at age 21 that goes to form the legal net value at the end of the year. Increase this $251,325 by 4 percent and we have $261,378, 24 : which is the net single premium necessary at age 22 to insure $1000 at the end of any year in which the policyholder may die. The part of the net single premium at age 22 that will pay net cost of insuring the amount at risk on this policy, during the year between age 22 and age 23, may be found in a similar manner, and so on, for each and every year to the table limit of age, at which time the legal net value will, when increased by 4 per cent, amount to $1000. This, too, after this policyholder has paid in advance each year the net cost of insuring the amount at risk on his policy during the year, and has also paid, out of the “ loading” added to the net premium, his proportion of the expenses, profits and contingencies on his policy every year from the time it was issued at age 20 until he has reached age 100. Having found that part of the legal net value that will, at the beginning of each year, pay net cost of insuring the amount at risk on this policy during that year, we can, by using the mortality table and rate of interest designated by law, find its value at age 20. For instance, we found above that $5,239 is that part of the legal net value of this policy at the end of the first year, that will pay net cost of insuring the amount at risk on this policy during the year between age 21 and age 22. To determine the net value at age 20 of $5,239, at age 21, in case the insured is then alive, we first find the present value of $1 in one year ; this at 4 per cent is Multiply yih by the number of persons shown by the table of mortality to be living at age 21, and divide the product by the number shown to be living at age 20, the result gives the net value at age 20 of $1 to be paid at age 21, in case the insured is then alive.* Mul- tiply this result by 5.239 and we have the net value at age 20 of the net cost of insuring the amount at risk on this policy during the second year. In a similar manner the net value at age 20 of the net cost of insuring the amount at risk on this policy each year to the table limit of age may be determined. The sum of all these yearly values gives the net present value at age 20 that will pay net cost of insuring the amount at risk on this policy every year to the table limit. * See Article VI. 25 V. W E have previously seen that, for an ordinary life policy of $1000, issued at age twenty, the legal net value at age sixty is $464,578. In this case the net annual premiums, paid and payable, are those called for by the Actuaries’ table of mortality and 4 per cent interest, viz., $12,948. Upon the same data the net annual premium necessary at age 60 to insure $1000 for life is $57,556. To obtain the legal net value of this policy, at age 60, we first found the difference between the net annual premium due to age 60* and that due to age 20 and then multiplied this difference by the value at age 60 of a life series of annual payments of $1 each. The same result would have been obtained by first finding the net single premium that will, at age 60, insure $1000 for life — (which is $599.433., see Table V) — and then subtracting from this net single premium, the value at age 60 of the net annual premiums yet to be paid. In case the company has contracted to furnish insurance for net premiums less than those called for by the table of mortality and rate of interest prescribed by the law, the legal net value required to be held for this policy by the company will be greater than that ob- tained above. For instance, suppose the net annual premiums the company has contracted to receive for this policy at and after age 60, are but $10 each — the value at age 60 of a life series of annual payments of $1 each is $10.4147 ; therefore, the value at the same age of a like series of annual payments of $10 each is $104,147. The net single premium — on the legal data above named, that will, at age 60, insure $1000 for life — being, as before stated, $599,433, we have $599,433 less $io4.i47=$495.286=the legal net value of this policy at age 60, in case the net annual premiums yet to be paid are $10 each. When these premiums were $12,948 each, the legal net value at age 60 was, as we have seen, $464,578. Voluminous tables, showing the net value of various kinds of 20 life policies at the end of each year of the policy, have been prepared and published at great expense. “ Valuation Tables” are very useful in finding the liability of a company on account of the accrued legal net value of the policies it has in force. These tables are constructed on the assumption that the net premiums receivable by the company are those called for by the table of mortality and rate of interest which were the bases upon which the calculations for the tables were made. When the net pre- miums the company has contracted to receive are less than those called for, as just stated, the tables show too small a net value. This must be corrected, in each such case, by adding to the amount given in the valuation tables, a sum equal to the value at the age for which the valuation is being made, of a series of payments each equal to the difference between the net premium called for by the data upon which the tables are computed, and the net premium the company has contracted to receive. If the company is to receive no further net premiums the legal net value is the net single premium necessary at that time to effect the insurance. If the future net premiums payable are greater than those called for by the State standard — only the value of that portion which is called for by that standard should be deducted from the net single premium in order to determine the legal net value. In this case that portion of the future net premiums which is in excess of that called for by the State standard must be treated as “loading” — it has no proper place in the princi- ples of net valuation, under the law. If the liability of a company, at any time, on account of the net value of the policies it has in force, is not accurately com- puted on the data prescribed by law, no proper conception can be formed of the legal condition of the company at that time. What the condition of the company may probably be at some future time is another question. The law requires of life insurance corporations something more than bare commercial solvency. It is not sufficient that probable future profits will enable a company to make up an existing present deficiency in the legal net value. In one or more of the States it is expressly provided that : 27 “ When the actual funds of any life insurance company doing business in this Commonwealth are not of a net value equal to its liabilities, counting as such the net value of its policies according to the prescribed table of mortality and rate of in- terest, it shall be the duty of the Insurance Commissioner to give notice to such company and its agents to discontinue issu- ing new policies within this Commonwealth until such time as its funds have become equal to its liabilities, valuing its policies as aforesaid ” The fund which the law designates as the “ net value ” of the policies a company has in force is usually called “ reserve.” In technical works on life insurance this fund is represented by the symbol H. The English writers who first used this symbol explain that it comes from the expression : “ How much must be in deposit?” In reference to this fund one of the leading actuaries says: u It does not belong to the company. It has been intrusted to them for a specific object, for which it ought to be sacredly reserved. “ The net valuation by which the legal reserve is calculated is proper and appropriate for many important purposes. “ It makes a just and proper report of the condition of the company, judged by the premiums, losses, and expenses already incurred. “ It tells the public if the company has not the legal reserve, that the managers have dissipated the whole of their capital, and that they ought not to be permitted to continue the busi- ness of insurance by making new contracts and issuing new policies.” The principles upon which calculations of legal net values are made not only enable us to determine the amount that should be held by the company to the credit of the policies it has in force, but these principles illustrate the fact that the net premium is composed of two parts — one of which pays each year the net cost of insuring the amount at risk on the policy during that year, and the other goes to form the legal net value that must be held by the company for the policy. It has already been shown that these separate parts of the net premium are susceptible of definite and easy computation, so that it may be known in advance, what portion of the policyholder’s payments will, on the legal data, be needed to insure the 28 amount at risk and what portion will go to form the “ reserve ” — as the companies style the fund which the law calls “ net value,” and which the older English writers considered to be a “ deposit” held to the credit of the policy. The “ loading,” added to net premiums for the purpose of providing for expenses, profits, contingencies, and so-called “ dividends ” to policyholders, is not directly taken into consid- eration in net valuations. The simple elementary principles upon which calculations of life insurance net values are based stand at the threshold of the business. Knowledge of these principles should be as common as that of calculating interest on money. Without clear conceptions on this subject, life in- surance cannot be understood any better than the business of banking can be comprehended by one who has no idea of the principles upon which calculations ot interest on money are based. Yet it has been recently said by the highest authority : “ It is wonderful what profound ignorance prevails in reference to the first elements of this subject.” Officers, trustees, and agents of companies, as a general rule, have made but little effort to have this ignorance abated ; on the contrary, they seem more than willing to have knowledge of these simple, elementary principles restricted to a few actuaries and consulting actuaries in the pay of the companies. With due deference to the opinions of those able business men who have the control of these corporations, it is believed that they make a great mistake in permitting knowledge of this subject to be confined to the “ initiated few.” A trust business, involving such immense amounts, cannot reasonably be ex- pected to permanently thrive and flourish upon the ignorance of the people. I'D VI. HE method by which we determine the sum that will, at l the beginning of any year, be just sufficient, on the legal data, to pay net cost of insuring the amount at risk on the policy during that year, has been explained. It is now pro- posed to illustrate in detail the principles used in computing the net present value of this sum. To do this — and at the same time give an example showing how subjects of this character were discussed one hundred years ago — the following extract is made from a work on Annuities, by Masseres, published in Lon- don in 1783. The first problem he gives is, “ To find the pre- sent value of a future sum of money which is certainly to be paid at the end of one or more years, according to any given rate of interest.” The rule by which this problem is solved has already been explained. The second problem given by Mas- seres is that to which attention is invited. He says : “ The doctrine of life annuities is by no means of so abstruse and difficult a nature as many people are apt to imagine. A moderate share of common sense, or capacity to reason justly, and a knowledge of common arithmetic, are all the qualities that are necessary to a right understanding of the principles on which it is founded.” He gives a table representing the probabilities of the duration of human life at the several ages therein mentioned, from the age of three years to the age of ninety-five, grounded on lists of the French Tontines or Long Annuities, and verified by a com- parison thereof with the necrologies, or mortuary registers, of several religious houses of both sexes, by M. de Parcieux : 30 Age. Persons Living. Age. Persons Living. Age. Persons Living. . Persons Age - Living. . Persons Age. Living. 3 IOOO 22 798 4 * 650 j 60 463 79 136 4 970 23 790 42 643 61 459 80 118 5 948 24 782 43 635 62 437 81 IOI 6 930 25 774 44 629 63 423 82 85 7 915 26 766 45 622 64 409 83 7 i 8 902 27 758 46 615 65 395 84 59 9 890 28 750 47 607 66 380 85 48 IO 880 29 742 48 599 i 6 7 364 86 38 ii 872 30 734 49 590 ; 68 347 87 29 12 866 31 726 50 58 i 69 329 88 22 13 860 32 718 5 i 57 i 70 310 89 16 14 854 33 710 52 560 7 i 291 90 11 15 848 34 702 53 549 72 271 9 i 7 16 842 35 694 54 538 73 251 92 4 1 7 a 35 36 686 55 526 74 231 93 2 18 828 37 678 56 5 i 4 ' 75 211 94 1 19 821 38 671 57 502 76 192 95 0 20 814 39 664 58 489 77 173 21 8l6 40 657 59 476 78 154 THE FUNDAMENTAL MAXIM OF THE DOCTRINE OF LIFE ANNUITIES. “ In every bargain between two persons concerning a grant of a sum of money to be paid by the one to the other at a given future time, in case the grantee or purchaser shall be then alive, the fair price of such future sum of money, a.ccord- ing to a given rate of the interest of money and a given table of the probabilities of the duration of human life, is to be ascer- tained in the following manner: We must suppose, in the first place, that the grantor of the future sum of money makes several hundred grants of the same kind, and upon exactly the same conditions, to as many different grantees, or purchasers, all of the same age with the first grantee ; and, in the second place, that these several purchasers die off in the interval between the time of making the grants and the time of payment, in the same proportion as persons of the same ages respectively are represented to do in the table of the prob- abilities of the duration of human life by which the calculation is to be governed, and, in the third place, we must suppose that the several sums of money paid by the several grantees of these future payments to the grantor of them as the price thereof, are improved by the said grantor, at compound interest, at the rate supposed in the question, during the whole interval of time be- tween the time of making the grants and the time at which the payments become due. And then we roust inquire what sum 31 each of the said grantees ought to pay to the grantor, to the end that, upon these three suppositions, he may, at the end of the said interval, or when the said payments become due, be neither a gainer nor a loser by the sum total of all his bargains, but be possessed of just enough money, arising from the sums form- erly paid him by the said grantees, to satisfy all the demands which will then be made upon him. And the sum which ought thus to be paid him by each of the said grantees, when he makes a great number of said grants to different persons, is the fair price which a single grantee ought to pay him for a grant for the said future sum of money, subject to the same conditions and contingencies when he makes only one such grant. This is a maxim which, I presume, will be admitted as self- evident, it being hardly possible to doubt of its truth. But if the reader should not admit it upon its own evidence, I confess I am unable to demonstrate it by means of any other proposi- tion more evident than itself. And, therefore, in this case, I must desire him to consider it as a definition of what is meant in the following pages by the expressions the ‘ fair f rice' or ‘ true value ’ of such a future contingent payment, since it is in that sense that the fair price or true value of such a future contingent payment can be collected from the table of the probability of the duration of human life above described.” PROBLEM II. “To find the sum of money which the purchaser of a future payment of one pound sterling, to be received at the end of any given number of years, provided the said purchaser shall then be living, ought to pay for it — the age of the said purchaser, and the rate of interest of money, and the probabilities of the duration of human life, being all given.” A SOLUTION OF THIS PROBLEM IN THE CASE OF A PAR- TICULAR EXAMPLE. Let the rate of interest of money be supposed to be three per cent, and the probabilities of the duration of human life such as they are represented to be in Monsieur de Parcieux’s table above mentioned ; and let the number of years at the end of which the said sum of one pound is to be paid to the grantee, or purchaser of it, if he be then alive, be thirty, and the age of the said grantee, or purchaser, twenty-five years. Then, in the first place, we must look into M. de Parcieux’s table to see how many persons of twenty-five years of age are there supposed to be all living at the same time. This number we shall find to be 774. We must therefore suppose that the 32 grantor of the one pound to the purchaser, proposed in the question, does not confine himself to that single grant, but makes 773 more such grants, of one pound each to as many different persons of the same age of twenty-five years, to be paid to them at the end of thirty years, or when they shall be fifty-five years old, if they shall then be living, but not to be paid to their executors, or other representatives, if they shall then be dead ; that is, we must suppose that he makes 774 such grants in all, including that of the purchaser proposed in the question. And we must likewise suppose that all these 774 purchasers have the same chance, one with the other, of living any given number of years, or that there is no apparent reason for supposing that any one of them is more likely to live to any given future age than any other. This done, we must inquire how many of these 774 purchasers of one pound each will be alive at the end of thirty years, supposing them to die off in the proportion mentioned in M. de Parcieux’s table. Now, it ap- pears by M. de Parcieux’s table, that out of 774 persons of the age of twenty-five years, all living at the same time, 52 6 will be alive at the age of fiftv-five years, or at the distance of thirty years. Therefore, out of the said 774 purchasers of these future payments of one pound, to be received at the end of thirty years, 526 will live to be entitled to them. Therefore, at the end of the said thirty years, the grantor of these future pay- ments will have 526 sums, of one pound each, to pay to the said surviving purchasers. And consequently, to the end that the said grantor may be neither a gainer nor a loser by the sum total of all his bargains, it is necessary that he should receive at the time of making the said grants 526 times the present value of one pound, payable at the end of thirty years, when the interest of money is three per cent, or 526 times the sum which, being improved continually at compound interest dur- ing the said term of thirty years of the said rate of interest, will at the end of that time amount to one pound ; because, in that case, if he improves the said sum (of 526 times the present value of one pound) so received, at compound interest, at the said rate of three per cent, during the whole thirty years, it will in that time increase to just 526 pounds, which is the sum he will then be obliged to pay to the surviving purchasers. The present value of one pound, payable at the end of thirty years, without being liable to any contingency, when the interest of money is three per cent, is .41198676 of a pound. Therefore, 526 times .41198676 of a pound, or £216.70503576, is the sum which the said grantor ought to receive, at the time of making the said grants, from all the 774 purchasers of them. There- fore, the sum which each of them ought then to pay him is the 33 774th part of £216 70503576, or .27998066 of a pound, or nearly .28 of a pound, or 55. 7 %d. And, consequently, when he makes only one such grant to a purchaser of twenty years of age, he ought to receive for it the same sum of .27998066 of a pound, or .28 of a pound, or 5 s. >]%d. I have solved the foregoing problem, in the case of a particu- lar example, for the sake of making the method of solution as clear and familiar as possible. But it is easy to see that the reasonings used in it extend to all other cases whatsoever, and, consequently, that the solution is really general.” The above principles apply to any table of mortality, to any rate of interest upon money, and to any unit of value — to one dollar just as well as to one pound sterling. Having found the net present value of one dollar to be paid at any designated future time, in case the insured person is then alive, the net present value of a similar payment of any other named sum becomes known. A right understanding of the principles upon which calcula- tions of life insurance legal net values are made — based upon a table of mortality and rate of interest designated by law — re- quires 44 a knowledge of common arithmetic ” and 44 a moderate capacity to reason justly.” The preposterous claim that great mathematical acquirement is essential to a clear understanding of the simple principles used in making the net calculations needed in ordinary life insurance is only equaled by the credu- lity of those who credit this absurd assumption