Quation.If the office insure a large number of persons (for the whole life or a term) at the premium p (the age being .r), what will be their accumulation in a years, upon the suppositions the working of which has been shown in the example already given ; it being sup posed that LA is insured to every one who dies Y The answer is in the following formula : for every person who, according to the tables, is alive at the end of the term of a years, there is remaining in the after all claims have been paid up to the end of the n years, and before the (n + 1)th premium is paid, the sum p 1) s(z+ n 1) } A fm(s) n) + a) As an example we shall verify the accumulations of 10 years in the The answer is, that the reserve of premiums for each person of the 5075 then remaining is 18-09002/., which for the whole is 91809'84/. The rough answer in the scheme worked out at length is 91,8091.
Now, this or 18/. Is. 10d., is what. is called the value of each man's policy at the expiration of the ten years; or the very utmost the office could afford to give him to surrender all claim, and to keep his future premiums to himself. But what is the nature of his claim on the office f Evidently this, that he Ins a right to make them give him a guarantee for the next ten years on payment of a premium of Ill. 12s. 34d., which could not be done at so low a rate for a new corner. Compute the premium for a person entering at 40, and insuring 1000/. for 10 years; or divide 1000 times 31(10)-3i(50) by N(39) N (49), and the answer will be found to be 13.800/., or 13/. ltts. If then any person aged 40, having been in the office ten years, were to put another person of the same age in his place at his own premium, he would obviously make that person a, present of the difference between 11.8148/. and 13.800L, at once (since a premium is to be paid imme diately), and for nine succeeding years, if the latter should live so long. And 13-800-11.0148 is while an annuity of 2'1852/. for nine yearn, at the ago of 40, is worth N(40)N(49) divided by D(40) end multiplied by 2.1552, or Add to this 2185/. for the imme diate difference, and we get 18.088/.., differing only a halfpenny from the sum which the office has in reserve. If then the person who is to take the place of the insured at 40 year of age, were to pay him an equil alent, he must, besides taking on himself the future pre miums, pay the retiring member which is therefore the value of the latter's policy. The last formula will always give the aceumula 1;on value of a policy, whether for the whole life or for a fixed term.
The preceding contains the most material calculations which are necessary in the management of an office, or rather, in forming an opinion on the management. of an office. It is to be remembered that all which has hitherto been said supposes the rates of mortality and interest. to be abeolntely known and invariable, the parties to enter on
their birthdays, and all charm to be adjusted at the terminations of whole years from the time of entry. We now proceed to the appli cation.
An aamrance company is a savings' bank, with a mutual tinder standing, presently to be noticed, between the contributors. To make out this proposition, let us suppose that A borrows money, and insures his life for the amount. as a security to his creditor. For this he has to pay a premium. If life were certain, the office of the company would be to receive and Invert thesepremiuma, which would be calcu. tared in such a manner as with their interest to amount to a sum suffi cient to discharge the loan In a settled time. At the end of this time the creditor (who has been all this while receiving Interest for his money from A) calls upon A to make hie claim upon the office, and repay the loan with the money received. If such an office existed, hfe being certain, the rationale of the proceeding would be that the creditor, though tolerably confident of A's power and willingness to make any yearly payment, whether of interest or instalment, will not trust him steadily to lay by and improve yearly instalments, but re quires that he should make his instalments payable to third parties, who are engaged not to return them on demand until they amount to a mum sufficient for the discharge of the debt. Stich an office certainly could not exist, on account of the uncertainty of individual life. So soon however as it is known that the duration of masses of individuals can be calculated with tolerable accuracy, there is a remedy for the individual uncertainties. Let a large number of debtors similarly situated with A, agree to be guarantees for one another ; that is, let. each of them pay during his life not only his own instalments, but inch additional sums as will provide the means of meeting the deficits of those who die, and the savings' tank thus constructed will become an manirance-office. Of course it matters nothing whether thaw debtors pay their instalments to a person agreed on among themselves, or go to a company which undertakes the management of such concerns. And again, it makes no difference whether the instalments be for liqui dation of debt, or to accumulate a provision for widow, and children. We have taken the MAO of debtors, because in such a case an office looks more like a mere indemnity-office than when its contributors enter for the benefit of their families; still however, in the former case, it is evident that the premiums are partly Instalments of debt, partly sums intended to make good the deficiency of the life-instal ments of those who die.