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RATE-MAKING JOINT-LIFE INSURANCE. Life insurance is sometimes conditional upon the fail ure of joint-lives and becomes payable upon the first death. Such are called joint-life insurances. They are sometimes taken by husband and wife in favor of each other or of their children. At one time joint-life insur ances of husband and wife, payable to the survivor, were somewhat common in the United States. More fre quently, however, these insurances have been taken by firms upon the joint lives of the partners, running to the surviving partners to aid in paying out the interest of the deceased partner.

For this reason, joint-life insurance is sometimes known as partnership insurance. That name, however, is not applied to it exclusively; separate insurances on the individual lives of the partners, if made payable to the other partners, are sometimes known likewise as part nership insurances.

The method of computing the net single premium for a whole life insurance of $1,000 has already been de scribed as follows : Add together the following present values : $1,000 discounted one year and multiplied by the probability of dying the first year; $1,000 discounted two years and multiplied by the probability of dying the sec ond year; and so on to the end of the table.

In like manner the method of finding the net single premium for a joint-life insurance of $1,000 is as fol lows : Add together the following present values : $1,000 discounted one year and multipled by the probability 73 that the joint lives i. e., at least one of them will fail the first year, and so on to the highest age in the mor tality table; when the oldest life reaches this age it must be assumed that it certainly fails, and, therefore, that the joint-lives certainly fail.

All that remains to be explained is the mode of find ing the probability that at least one life out of the joint lives will fail the first year, the second year, and so on. If we go to work to get this directly, it will become a very complicated matter. Thus, if there are only two lives, it evidently consists of three separate probabilities viz.: (I) that both lives will fail, (2) that the one life will fail and the other survive, and (3) that the other life will fail and the one survive. Where more lives are involved, the probability can be separated into a larger number of distinct, separate probabilities in an increas ing ratio for each life added.

But, remembering that certainty is unity and that all probabilities are fractions, we can very easily get the probability that at least one of any number of lives will fail; because it is certainty, less the chance that none of them will fail, which, in turn, is the chance that all will survive. And the probability that all will survive is the

product of the probability that one will survive and the probability that another will survive, and so on.

In computing the net single premium for a joint-life insurance for a certain number of years only, all that is necessary is to add together the present values computed in accordance with the foregoing for that number of years only.

To determine the net single premium for a joint-life endowment insurance, add the net single premium for a joint-life insurance and the net single premium of a joint-life pure endowment, both for the desired term.

A joint-life pure endowment is a promise to pay a sum 74 of money at the end of the term if all the lives involved have survived. The net single premium for a joint-life pure endowment is the amount of the endowment dis counted for the term of years and multiplied by the probability that all will survive the term, which, as has been explained, is the product of the probabilities that each will severally survive the term.

The present value of a joint-life annuity is the sum of the net single premiums for joint-life pure endowments of $i due in one, two, three, etc., years up to the number of years by which the present age of the oldest life falls short of the highest age in the table. A joint-life an nuity due is a joint-life annuity with one payment imme diate, and its present value is, therefore, greater by the amount of the annuity payment made in advance.

The present value of a joint-life annuity for a term of years is the sum of the net single premiums for joint-life pure endowments of $i due in one, two, three, four, etc., years up to the number of years in the term.

The present value of a joint-life annuity due for a term of years is the present value of a joint-life annuity for a term one year shorter plus $1.

The net annual premium of a joint-life insurance is found by dividing the net single premium by the present value of a joint-life annuity due.

The net annual premium, limited to a term of years, for a joint-life insurance, is found by dividing the net single premium by the present value of a joint-life an nuity due for the payment term.

The net annual premium for a joint-life insurance for a certain term, or of a joint-life endowment insurance for a certain term, is found by dividing the respective net sin gle premium by the present value of a joint-life annuity due for the same term.