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# Rate-Making-Endowment Insurance

RATE-MAKING-ENDOWMENT INSURANCE. Endowment insurances are often called endowments, but that name is more appropriately applied to pure en dowments. An endowment insurance is payable to the insured at the end of a fixed period if he survives, or to the beneficiary if he dies within the period.

It will readily be seen, then, that an endowment insur ance consists of two things combined, viz. : A life insur ance for the term with no return on survival, and a pure endowment on survival of the term with no return in event of prior death. Were each made a separate contract, one would pay in event of death and the other in event of survival.

To illustrate, the following is the method for finding the net single premium for an endowment insurance of \$1,000 issued at age 90, payable at the end of five years or at prior death.

\$1,124,220 Dividing by 1,319, we get \$852.25 as the net single pre mium for the five years' insurance of \$1,000 from age.

At the end of the five years, in addition to paying \$95,000 to the beneficiaries of those who die during that year, there must be \$89,000 paid to the 89 survivors, the discounted value of which is \$73,151.50. Dividing this by 1,319, we have \$55.46 as the net single premium at age 90 for a pure endowment of \$1,00o due in five years.

Adding together the net single premiums for the insur ance and for the pure endowment we have \$852.25 + = \$907.71, as the net single premium for a five year endowment insurance of \$1,000 issued at age 9o.

Or we might have reasoned thus : Since \$1,124,220 is the discounted value of all the insurances, and \$73,151.50 the discounted value of all the endowments, \$1,124,220 + \$73,151.50 = \$1,197,371.50 is the discounted value of the 1,319 endowment insurances of \$1,0oo each, whence \$1,197,371.50 1,319 = \$907.71, the net single premium for an endowment insurance of \$i,000 at age 90.

We have seen that \$2.398 is the present value at age 90 of a life annuity due of \$1 limited to five years. The net annual premium for a five-year endowment insurance for \$1,000, issued at age 90, is a life annuity due, at age 90, limited to five years, and the problem is to determine the amount of it. Its present value must be equal to the net single premium of such an endowment insurance, which is \$907.71. Therefore, the net annual premium is as many dollars as \$2.398 is contained in \$907.71, which is \$378.53.

Instead of thus deriving the net annual premium for the endowment insurance from the single premium for the same, and the latter from the net single premiums for the temporary insurance and the pure endowment, we might get the net annual premium for the endowment in surance by first computing the net annual premiums for the temporary insurance and for the pure endowment separately, and then adding these together to make the net annual premium for the endowment insurance.

Semi-endowment insurances provide for the payment of a certain sum in event of death during a certain period, or of one-half as much on survival. Thus, for a semi-endowment insurance for five years at age 9o, of \$i,000, we would combine the net single pre miums for \$i,000 insurance for five years and for \$5oo pure ,endowment in five years to get the net single pre mium for the semi-endowment insurance, thus : \$852.25 + \$27.73 = \$879.98; and in like manner for the net annual premium or by dividing this net single premium by the present value of a life annuity due of \$1 at age 90 limited to five years.

A double endowment insurance means an insurance for a certain sum in event of death during a certain period, and a pure endowment for double that sum payable in event of survival of the period. The mode of arriving at net single and annual premiums is identical with that already described.

Any policy which, at the end of a given period, pro duces a certain accumulation upon survival, besides car rying an insurance during the period, may be considered a partial endowment insurance, and the net single and annual premiums may be computed in accordance with the foregoing principles.